Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1817 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)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,23,28)(19,24,31,38)(20,25,22,27)(21,26,34,41)(29,36,35,42)(30,37,45,48)(32,39,33,40)(43,46,44,47);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(19,29)(20,22)(21,32)(24,36)(25,27)(26,39)(31,35)(33,34)(38,42)(40,41)(43,44)(46,47);(9,12,13,11)(10,14,15,16)(17,19,23,31)(18,24,28,38)(20,29,22,35)(21,30,34,45)(25,36,27,42)(26,37,41,48)(32,43,33,44)(39,46,40,47);(2,6)(5,7)(17,20,23,22)(18,25,28,27)(19,29,31,35)(21,33,34,32)(24,36,38,42)(26,40,41,39)(30,44,45,43)(37,47,48,46);(3,8)(5,7)(17,21,23,34)(18,26,28,41)(19,30,31,45)(20,32,22,33)(24,37,38,48)(25,39,27,40)(29,43,35,44)(36,46,42,47)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6597711);