Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#466 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #3573 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,9)(3,16)(5,10)(6,7)(11,34)(12,28)(13,30)(14,29)(15,31)(17,42)(18,45)(19,26)(21,33)(22,23)(24,25)(27,36)(35,57)(37,55)(38,52)(39,54)(40,53)(41,51)(43,48)(44,61)(46,47)(50,59)(58,64)(60,63);(1,2,10,30,7,14,25,31,8,15,24,29,6,13,5,9)(3,11,32,34,16,36,45,51,19,37,49,55,26,41,18,27)(4,12,33,54,23,40,46,52,20,38,47,53,22,39,21,28)(17,35,56,57,42,59,61,63,43,58,62,64,48,60,44,50);(1,3)(2,11)(4,17)(5,18)(6,26)(7,16)(8,19)(9,27)(10,32)(12,35)(13,41)(14,36)(15,37)(20,43)(21,44)(22,48)(23,42)(24,49)(25,45)(28,50)(29,55)(30,34)(31,51)(33,56)(38,58)(39,60)(40,59)(46,61)(47,62)(52,63)(53,64)(54,57);(1,4)(2,12)(3,17)(5,21)(6,22)(7,23)(8,20)(9,28)(10,33)(11,35)(13,39)(14,40)(15,38)(16,42)(18,44)(19,43)(24,47)(25,46)(26,48)(27,50)(29,53)(30,54)(31,52)(32,56)(34,57)(36,59)(37,58)(41,60)(45,61)(49,62)(51,63)(55,64);(4,20)(12,38)(17,43)(21,46)(22,23)(28,52)(33,47)(35,58)(39,40)(42,48)(44,61)(50,63)(53,54)(56,62)(57,64)(59,60)Polycyclic presentation:
2^9|b^2=fh,f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,h^a=hi,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530188);