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)(17,18,21,35,22,26,23,27)(19,28,36,40,30,24,31,39)(20,32,37,25,34,42,33,41)(29,44,46,43,45,48,47,38);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,28)(27,40)(32,48)(33,47)(34,45)(35,39)(37,46)(41,43)(42,44);(10,13)(14,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,38)(26,41)(27,42)(28,43)(30,45)(31,46)(32,35)(36,47)(39,48)(40,44);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(18,26)(19,30)(25,41)(27,35)(29,45)(31,36)(32,42)(46,47);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,21,22,23)(18,27,26,35)(19,31,30,36)(20,33,34,37)(24,39,28,40)(25,42,41,32)(29,46,45,47)(38,48,43,44)Polycyclic presentation:
2^9|a^2=fg,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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, 6450550);