Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1665 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)(3,8)(4,6)(5,7)(17,18)(19,24)(20,31)(21,26)(22,37)(23,27)(25,33)(28,45)(29,39)(30,40)(32,48)(34,43)(35,41)(36,44)(38,47)(42,46);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,39)(20,32,34,28)(21,35,22,30)(25,42,44,38)(26,41,37,40)(31,48,43,45)(33,46,36,47);(10,13)(14,16)(17,20,22,33,23,34,21,36)(18,25,26,31,27,44,37,43)(19,28,30,47,29,32,35,46)(24,38,41,45,39,42,40,48);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,29)(24,39)(28,32)(30,35)(38,42)(40,41)(45,48)(46,47);(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,30,29,35)(20,33,34,36)(24,40,39,41)(25,43,44,31)(28,46,32,47)(38,45,42,48)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^c=ei,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6465778);