Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907869 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,28)(20,32)(21,27)(22,26)(23,37)(24,30)(25,35)(29,43)(31,40)(33,47)(34,41)(36,39)(38,48)(42,44)(45,46);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,28)(27,40)(32,46)(33,45)(34,48)(35,44)(37,39)(41,42)(43,47);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,21,34,22,33,23,35)(18,25,37,47,26,41,27,32)(19,29,36,48,30,45,31,44)(24,38,39,43,28,42,40,46);(2,6)(5,7)(11,16)(13,15)(18,26)(19,30)(20,33)(27,37)(31,36)(34,35)(38,42)(43,46);(10,14)(13,15)(17,21,22,23)(18,27,26,37)(19,31,30,36)(20,34,33,35)(24,39,28,40)(25,32,41,47)(29,44,45,48)(38,43,42,46)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6816386);