Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(17,18,21,26,22,27,23,28)(19,29,36,39,31,24,32,41)(20,33,37,25,35,42,34,40)(30,44,46,43,45,48,47,38);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,41)(27,29)(28,39)(33,48)(34,47)(35,45)(37,46)(40,43)(42,44);(10,13)(14,16)(17,20)(18,25)(19,30)(21,34)(22,35)(23,37)(24,38)(26,33)(27,40)(28,42)(29,43)(31,45)(32,46)(36,47)(39,44)(41,48);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(19,31)(24,29)(30,45)(32,36)(38,43)(39,41)(44,48)(46,47);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,21,22,23)(18,26,27,28)(19,32,31,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,46,45,47)(38,44,43,48)Polycyclic presentation:
2^9|a^2=fg,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6450526);