Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511763 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32878 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #63093 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3573 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3055 conjugacy classes of subgroups; only the 220 normal subgroups are shown.
This group has 3055 conjugacy classes of subgroups — too many to draw. Only its 220 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:
(1,2,8,14)(3,15,20,54)(4,10,23,42)(5,11,26,45)(6,28,32,51)(7,33,30,13)(9,35,39,18)(12,47,49,31)(16,52,57,102)(17,53,60,103)(19,62,34,48)(21,63,65,40)(22,67,70,99)(24,72,69,43)(25,74,77,100)(27,79,76,46)(29,50,81,38)(36,82,85,56)(37,83,87,59)(41,91,93,71)(44,95,97,78)(55,104,106,101)(58,108,73,92)(61,109,80,96)(64,113,116,119)(66,90,115,112)(68,94,120,86)(75,98,121,88)(84,105,110,122)(89,117,111,126)(107,124,118,127)(114,123,128,125);(1,3,8,20)(2,9,14,39)(4,16,23,57)(5,17,26,60)(6,29,32,81)(7,34,30,19)(10,36,42,85)(11,37,45,87)(12,48,49,62)(13,50,33,38)(15,51,54,28)(18,47,35,31)(21,55,65,106)(22,68,70,120)(24,73,69,58)(25,75,77,121)(27,80,76,61)(40,84,63,110)(41,92,93,108)(43,94,72,86)(44,96,97,109)(46,98,79,88)(52,99,102,67)(53,100,103,74)(56,91,82,71)(59,95,83,78)(64,114,116,128)(66,118,115,107)(89,124,111,127)(90,125,112,123)(101,113,104,119)(105,126,122,117);(1,4)(2,10)(3,16)(5,21)(6,22)(7,24)(8,23)(9,36)(11,40)(12,41)(13,43)(14,42)(15,52)(17,55)(18,56)(19,58)(20,57)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,71)(32,70)(33,72)(34,73)(35,82)(37,84)(38,86)(39,85)(44,89)(45,63)(46,90)(47,91)(48,92)(49,93)(50,94)(51,99)(53,101)(54,102)(59,105)(60,106)(61,107)(62,108)(74,119)(75,114)(76,115)(77,116)(78,117)(79,112)(80,118)(81,120)(83,122)(87,110)(88,123)(95,126)(96,124)(97,111)(98,125)(100,113)(103,104)(109,127)(121,128);(1,5)(2,11)(3,17)(4,21)(6,25)(7,27)(8,26)(9,37)(10,40)(12,44)(13,46)(14,45)(15,53)(16,55)(18,59)(19,61)(20,60)(22,64)(23,65)(24,66)(28,74)(29,75)(30,76)(31,78)(32,77)(33,79)(34,80)(35,83)(36,84)(38,88)(39,87)(41,89)(42,63)(43,90)(47,95)(48,96)(49,97)(50,98)(51,100)(52,101)(54,103)(56,105)(57,106)(58,107)(62,109)(67,119)(68,114)(69,115)(70,116)(71,117)(72,112)(73,118)(81,121)(82,122)(85,110)(86,123)(91,126)(92,124)(93,111)(94,125)(99,113)(102,104)(108,127)(120,128)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,e^b=eh,h^b=hi,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418962);