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 | #24425 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(17,18,22,27)(19,28,30,24)(20,33,35,40)(21,26,23,36)(25,37,41,34)(29,44,45,48)(31,43,32,39)(38,46,42,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,27,28)(20,29,35,45)(21,31,23,32)(25,38,41,42)(26,39,36,43)(33,48,40,44)(34,46,37,47);(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,37)(24,38)(26,40)(27,41)(28,42)(30,45)(31,46)(32,47)(33,36)(39,44)(43,48);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,30)(24,28)(29,45)(31,32)(38,42)(39,43)(44,48)(46,47);(11,15)(14,16)(17,21,22,23)(18,26,27,36)(19,31,30,32)(20,34,35,37)(24,39,28,43)(25,40,41,33)(29,46,45,47)(38,44,42,48)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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, 6444807);