Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814805 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61337 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18)(19,23)(20,25)(21,24)(22,27)(26,35)(28,39)(29,37)(30,38)(31,36)(32,41)(33,42)(34,40)(43,46)(44,48)(45,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,23)(20,32)(21,28)(22,31)(24,39)(25,41)(26,38)(27,36)(29,34)(30,35)(33,44)(37,40)(42,48)(43,47)(45,46);(18,24,27,26)(19,28,31,30)(23,36)(25,40,42,43)(29,44,45,32)(37,47)(38,39)(41,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(20,33)(25,42)(29,45)(32,44)(34,46)(37,47)(40,43)(41,48);(3,11)(6,9)(10,16)(14,15)(17,20,21,34,22,33,35,46)(18,25,24,40,27,42,26,43)(19,29,30,32,31,45,28,44)(23,37,38,41,36,47,39,48)Polycyclic presentation:
2^9|a^2=f,c^2=ghi,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722544);