Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #31037 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,19,27,22,26,30,23)(20,31,32,40,34,38,28,24)(21,35,29,37,36,25,41,42)(33,46,45,39,47,44,43,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,29,36,41)(24,38,40,31)(25,37,35,42)(33,45,47,43)(39,46,48,44);(10,14)(15,16)(17,20,22,34)(18,24,26,40)(19,28,30,32)(21,33,36,47)(23,31,27,38)(25,39,35,48)(29,43,41,45)(37,44,42,46);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,36)(18,25,26,35)(19,29,30,41)(20,33,34,47)(23,37,27,42)(24,39,40,48)(28,43,32,45)(31,44,38,46);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(21,36)(25,35)(29,41)(33,47)(37,42)(39,48)(43,45)(44,46)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6775106);