Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501024 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24354 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #62637 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
957 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 957 conjugacy classes of subgroups — too many to draw. Only its 160 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:
(1,2,9,39,8,13,6,14)(3,16,33,43,21,10,19,53)(4,22,34,11,25,54,26,48)(5,28,31,49,32,12,29,51)(7,15,40,93,38,52,35,55)(17,61,70,58,64,95,65,41)(18,42,68,98,69,59,66,96)(20,60,89,56,72,44,36,91)(23,73,81,101,76,99,71,45)(24,74,80,82,77,102,78,46)(27,57,90,47,62,108,37,92)(30,83,87,104,88,50,84,107)(63,111,115,94,113,127,114,109)(67,97,85,105,118,110,116,125)(75,119,117,100,120,128,121,123)(79,106,86,124,112,126,122,103);(1,3,8,21)(2,10,13,16)(4,23,25,76)(5,18,32,69)(6,33,9,19)(7,36,38,89)(11,45,48,101)(12,42,28,59)(14,53,39,43)(15,56,52,91)(17,62,64,27)(20,40,72,35)(22,73,54,99)(24,75,77,120)(26,81,34,71)(29,68,31,66)(30,85,88,116)(37,70,90,65)(41,92,58,47)(44,93,60,55)(46,100,82,123)(49,98,51,96)(50,105,83,125)(57,61,108,95)(63,112,113,79)(67,84,118,87)(74,119,102,128)(78,117,80,121)(86,114,122,115)(94,124,109,103)(97,104,110,107)(106,111,126,127);(1,4)(2,11)(3,17)(5,24)(6,34)(7,37)(8,25)(9,26)(10,41)(12,46)(13,48)(14,54)(15,57)(16,58)(18,63)(19,70)(20,71)(21,64)(22,39)(23,36)(27,35)(28,82)(29,80)(30,86)(31,78)(32,77)(33,65)(38,90)(40,62)(42,94)(43,61)(44,99)(45,56)(47,55)(49,74)(50,106)(51,102)(52,108)(53,95)(59,109)(60,73)(66,115)(67,117)(68,114)(69,113)(72,81)(75,85)(76,89)(79,87)(83,126)(84,112)(88,122)(91,101)(92,93)(96,127)(97,128)(98,111)(100,105)(103,107)(104,124)(110,119)(116,120)(118,121)(123,125);(1,5)(2,12)(3,18)(4,24)(6,31)(7,30)(8,32)(9,29)(10,42)(11,46)(13,28)(14,51)(15,50)(16,59)(17,63)(19,68)(20,67)(21,69)(22,74)(23,75)(25,77)(26,80)(27,79)(33,66)(34,78)(35,87)(36,85)(37,86)(38,88)(39,49)(40,84)(41,94)(43,98)(44,97)(45,100)(47,103)(48,82)(52,83)(53,96)(54,102)(55,107)(56,105)(57,106)(58,109)(60,110)(61,111)(62,112)(64,113)(65,115)(70,114)(71,117)(72,118)(73,119)(76,120)(81,121)(89,116)(90,122)(91,125)(92,124)(93,104)(95,127)(99,128)(101,123)(108,126)Polycyclic presentation:
2^9|a^2=fi,b^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408223);