Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,28,21,27)(19,25,35,37,30,40,31,41)(20,32,36,42,34,26,33,38)(29,43,46,48,45,39,44,47);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(18,24)(20,33)(21,23)(25,37)(27,28)(29,44)(31,35)(34,36)(38,42)(40,41)(45,46)(47,48);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,29)(21,31)(22,30)(23,35)(24,37)(26,39)(27,41)(28,40)(32,43)(33,44)(34,45)(36,46)(38,47)(42,48);(11,16)(13,15)(17,20,21,33,22,34,23,36)(18,26,27,42,28,32,24,38)(19,29,31,44,30,45,35,46)(25,39,41,48,40,43,37,47);(10,14)(13,15)(19,30)(25,40)(29,45)(31,35)(37,41)(39,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6926316);