Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | |
1459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(17,18)(19,28)(20,24)(21,25)(22,27)(23,31)(26,29)(30,42)(32,37)(33,39)(34,40)(35,41)(36,46)(38,43)(44,48)(45,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(20,33)(24,39)(30,44)(34,35)(36,47)(40,41)(42,48)(45,46);(1,3)(2,7)(4,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,29,43,22,32,21,31)(18,23,25,37,27,38,26,28)(20,30,34,45,33,44,35,46)(24,36,41,48,39,47,40,42);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(17,20,22,33)(18,24,27,39)(19,30,32,44)(21,35,29,34)(23,36,38,47)(25,41,26,40)(28,42,37,48)(31,46,43,45);(2,6)(7,8)(10,14)(13,15)(17,21,22,29)(18,25,27,26)(19,31,32,43)(20,34,33,35)(23,37,38,28)(24,40,39,41)(30,45,44,46)(36,42,47,48)Polycyclic presentation:
2^9|c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6818028);