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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,31,41)(20,25,35,44)(21,26,36,22)(28,39,46,47)(29,30,40,37)(32,42,45,48)(33,34,43,38);(1,2)(3,6)(4,8)(5,7)(9,11,12,15)(10,13,14,16)(17,19,23,31)(18,24,27,41)(20,32,35,45)(21,29,36,40)(22,37,26,30)(25,42,44,48)(28,38,46,34)(33,47,43,39);(10,14)(13,16)(17,20,23,35)(18,25,27,44)(19,28,31,46)(21,33,36,43)(22,38,26,34)(24,39,41,47)(29,42,40,48)(30,45,37,32);(1,3,5,4)(2,6,7,8)(17,21)(18,26)(19,29)(20,33)(22,27)(23,36)(24,30)(25,34)(28,42)(31,40)(32,47)(35,43)(37,41)(38,44)(39,45)(46,48);(1,4,5,3)(2,6,7,8)(19,30)(21,36)(22,26)(24,40)(28,45)(29,41)(31,37)(32,46)(33,43)(34,38)(39,48)(42,47)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=i,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,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, 6588732);