Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,30,37)(20,24,31,38)(21,25,33,40)(27,34,42,46)(28,35,29,36)(32,39,44,48)(41,45,43,47);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19,22,30)(18,23,26,37)(20,27,31,42)(21,28,33,29)(24,34,38,46)(25,35,40,36)(32,43,44,41)(39,47,48,45);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,22,31)(18,24,26,38)(19,27,30,42)(21,32,33,44)(23,34,37,46)(25,39,40,48)(28,43,29,41)(35,47,36,45);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(19,28)(20,31)(21,33)(23,35)(24,38)(25,40)(27,41)(29,30)(34,45)(36,37)(42,43)(46,47);(2,6)(7,8)(10,14)(13,15)(20,31)(24,38)(27,42)(32,44)(34,46)(39,48)(41,43)(45,47)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=fgi,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7476856);