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)(17,18,22,27)(19,23,30,26)(20,31,33,36)(21,25,35,42)(24,38,40,28)(29,37,34,41)(32,44,46,47)(39,45,48,43);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,26)(20,28,33,38)(21,34,35,29)(24,36,40,31)(25,41,42,37)(32,45,46,43)(39,44,48,47);(1,3)(2,7)(4,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,30,38,22,33,19,28)(18,24,23,36,27,40,26,31)(21,32,34,45,35,46,29,43)(25,39,37,47,42,48,41,44);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(17,21,22,35)(18,25,27,42)(19,29,30,34)(20,32,33,46)(23,37,26,41)(24,39,40,48)(28,43,38,45)(31,44,36,47);(2,6)(7,8)(10,14)(13,15)(21,35)(25,42)(29,34)(32,46)(37,41)(39,48)(43,45)(44,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fi,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, 6818051);