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 | #36018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1794 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)(21,33)(26,42)(30,45)(32,35)(34,41)(39,48)(43,47)(44,46);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,18)(19,28)(20,25)(21,34)(22,36)(23,27)(24,29)(26,32)(30,39)(31,38)(33,41)(35,42)(37,40)(43,46)(44,47)(45,48);(9,12,13,11)(10,14,15,16)(17,19)(18,24)(20,29)(21,30)(22,37)(23,31)(25,38)(26,39)(27,40)(28,36)(32,44)(33,45)(34,43)(35,46)(41,47)(42,48);(2,6)(5,7)(17,20,23,22)(18,25,27,36)(19,29,31,37)(21,35,33,32)(24,38,40,28)(26,34,42,41)(30,46,45,44)(39,43,48,47);(3,8)(5,7)(17,21,23,33)(18,26,27,42)(19,30,31,45)(20,32,22,35)(24,39,40,48)(25,41,36,34)(28,43,38,47)(29,44,37,46)Polycyclic presentation:
2^9|b^2=fg,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,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, 6597746);