Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #36085 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2236 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)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,32,41)(20,25,35,43)(21,26,37,22)(28,39,46,47)(29,31,40,30)(33,42,45,48)(34,36,44,38);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,33)(21,29)(22,30)(23,32)(25,42)(26,31)(27,41)(28,36)(34,47)(35,45)(37,40)(38,46)(39,44)(43,48);(10,14)(13,16)(17,20,23,35)(18,25,27,43)(19,28,32,46)(21,34,37,44)(22,38,26,36)(24,39,41,47)(29,42,40,48)(30,33,31,45);(2,6)(5,7)(17,21)(18,26)(19,29)(20,34)(22,27)(23,37)(24,31)(25,36)(28,42)(30,41)(32,40)(33,47)(35,44)(38,43)(39,45)(46,48);(3,8)(5,7)(19,30)(20,35)(21,37)(22,26)(24,29)(25,43)(28,45)(31,32)(33,46)(39,48)(40,41)(42,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6585140);