Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 | #36020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1782 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)(19,23)(20,25)(21,24)(22,26)(27,42)(28,36)(29,37)(30,35)(31,40)(32,38)(33,41)(34,39)(43,47)(44,46)(45,48);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19,21,29)(18,23,24,37)(20,31,32,28)(22,27,34,30)(25,40,38,36)(26,42,39,35)(33,44,45,43)(41,46,48,47);(9,12,13,11)(10,14,15,16)(18,24)(19,27)(22,34)(23,35)(25,38)(28,43)(29,30)(31,44)(33,45)(36,46)(37,42)(40,47);(2,6)(5,7)(17,20)(18,25)(19,28)(21,32)(22,33)(23,36)(24,38)(26,41)(27,43)(29,31)(30,44)(34,45)(35,46)(37,40)(39,48)(42,47);(3,8)(5,7)(18,24)(19,27)(20,32)(22,34)(23,35)(28,44)(29,30)(31,43)(36,47)(37,42)(40,46)(41,48)Polycyclic presentation:
2^9|b^2=fg,c^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6597774);