Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499905 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23918 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3573 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1317 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 1317 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2)(3,17)(4,23)(5,30)(6,15)(7,37)(8,13)(9,16)(10,20)(11,27)(12,31)(14,39)(18,59)(19,42)(21,69)(22,44)(24,74)(25,76)(26,81)(28,54)(29,73)(32,87)(33,51)(34,52)(35,45)(36,55)(38,56)(40,53)(41,93)(43,71)(46,62)(47,104)(48,82)(49,83)(50,90)(57,64)(58,68)(60,111)(61,95)(63,101)(65,109)(66,98)(67,100)(70,97)(72,77)(75,110)(78,84)(79,120)(80,107)(85,86)(88,106)(89,108)(91,102)(92,96)(94,112)(99,117)(103,116)(105,122)(113,121)(114,128)(115,119)(118,127)(123,124)(125,126);(1,3)(2,10)(4,24)(5,19)(6,35)(7,21)(8,20)(9,22)(11,46)(12,42)(13,44)(14,43)(15,17)(16,45)(18,28)(23,59)(25,75)(26,63)(27,70)(29,61)(30,57)(31,66)(32,65)(33,64)(34,67)(36,91)(37,58)(38,92)(39,68)(40,71)(41,48)(47,103)(49,96)(50,99)(51,100)(52,98)(53,102)(54,95)(55,69)(56,97)(60,77)(62,82)(72,110)(73,101)(74,81)(76,121)(78,116)(79,113)(80,114)(83,93)(84,123)(85,109)(86,117)(87,119)(88,124)(89,115)(90,118)(94,105)(104,112)(106,126)(107,111)(108,127)(120,128)(122,125);(1,4)(2,11)(3,18)(5,25)(6,26)(7,38)(8,28)(9,29)(10,41)(12,47)(13,48)(14,54)(15,56)(16,49)(17,46)(19,60)(20,61)(21,70)(22,63)(23,37)(24,35)(27,39)(30,84)(31,77)(32,88)(33,79)(34,80)(36,83)(40,82)(42,94)(43,101)(44,96)(45,97)(50,107)(51,106)(52,105)(53,73)(55,81)(57,103)(58,95)(59,69)(62,68)(64,75)(65,116)(66,114)(67,113)(71,93)(72,85)(74,102)(76,87)(78,86)(89,122)(90,104)(91,92)(98,126)(99,128)(100,123)(108,120)(109,111)(110,119)(112,117)(115,124)(118,125)(121,127);(1,5,9,34)(2,12,16,51)(3,19,22,67)(4,25,29,80)(6,33,8,31)(7,32,40,90)(10,42,45,100)(11,47,49,106)(13,52,15,30)(14,50,55,87)(17,57,44,98)(18,60,63,113)(20,66,35,64)(21,65,71,118)(23,72,73,120)(24,75,61,114)(26,79,28,77)(27,78,83,122)(36,89,39,86)(37,85,53,108)(38,88,82,104)(41,94,97,123)(43,99,69,119)(46,103,96,126)(48,105,56,84)(54,107,81,76)(58,109,102,127)(59,110,101,128)(62,112,92,124)(68,117,91,115)(70,116,93,125)(74,121,95,111)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,f^c=fh,g^c=gi,h^c=hi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407102);