Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #502201 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25148 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,7,32)(3,15,19,49,20,50,33,45)(4,21,23,41,24,40,31,10)(5,25,28,46,30,59,29,12)(9,26,36,57,37,61,38,18)(11,27,42,58,44,62,43,22)(16,51,53,64,17,35,54,47)(34,52,63,56,39,55,48,60);(1,3,6,19,8,20,7,33)(2,9,13,36,14,37,32,38)(4,17,23,54,24,16,31,53)(5,26,28,57,30,61,29,18)(10,39,21,48,41,34,40,63)(11,35,42,47,44,51,43,64)(12,45,25,15,46,49,59,50)(22,56,27,55,58,60,62,52);(1,4)(2,10)(3,16)(5,22)(6,31)(7,23)(8,24)(9,34)(11,12)(13,40)(14,41)(15,47)(17,20)(18,52)(19,54)(21,32)(25,43)(26,60)(27,29)(28,62)(30,58)(33,53)(35,49)(36,48)(37,39)(38,63)(42,59)(44,46)(45,51)(50,64)(55,57)(56,61);(1,4)(2,11)(3,17)(5,27)(6,23)(7,31)(8,24)(9,35)(10,12)(13,42)(14,44)(15,48)(16,20)(18,56)(19,54)(21,25)(22,29)(26,55)(28,58)(30,62)(32,43)(33,53)(34,49)(36,47)(37,51)(38,64)(39,45)(40,59)(41,46)(50,63)(52,61)(57,60)Polycyclic presentation:
2^9|a^2=f,b^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,f^c=fi,e^d=eh,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409397);