Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1660 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)(17,18,22,28)(19,29,30,24)(20,34,37,26)(21,27,23,25)(31,40,35,43)(32,41,33,39)(36,44,38,42)(45,48,46,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19)(18,24)(20,35)(21,32)(22,30)(23,33)(25,41)(26,43)(27,39)(28,29)(31,37)(34,40)(36,46)(38,45)(42,47)(44,48);(10,13)(14,15)(18,25)(19,30)(21,23)(24,39)(26,42)(27,28)(29,41)(31,35)(34,44)(36,38)(40,47)(43,48);(2,6)(5,7)(17,20,22,37)(18,26,28,34)(19,31,30,35)(21,38,23,36)(24,40,29,43)(25,42,27,44)(32,46,33,45)(39,47,41,48);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,27,28,25)(19,32,30,33)(20,36,37,38)(24,39,29,41)(26,42,34,44)(31,45,35,46)(40,48,43,47)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6698642);