Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814811 |
| 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 | #3045 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1563 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)(17,18,23,29)(19,24,32,38)(20,33,35,45)(21,26,39,31)(22,28,40,27)(25,42,43,37)(30,36,46,48)(34,47,44,41);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,34)(21,27)(22,26)(23,32)(25,36)(28,39)(29,38)(30,37)(31,40)(33,47)(35,44)(41,45)(42,46)(43,48);(17,20,23,35)(18,25)(19,30)(21,36,39,48)(22,37,40,42)(24,41,38,47)(26,44)(27,33)(28,45)(29,43)(31,34)(32,46);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(20,35)(25,43)(30,46)(33,45)(34,44)(36,48)(37,42)(41,47);(3,11)(6,9)(10,16)(14,15)(17,21,22,38,23,39,40,24)(18,26,28,19,29,31,27,32)(20,36,37,47,35,48,42,41)(25,44,45,30,43,34,33,46)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,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,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722535);