Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806827 |
| 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 | #3494 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1571 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,26)(20,33)(21,35)(22,27)(23,39)(24,29)(25,31)(28,36)(30,40)(32,41)(34,43)(37,44)(38,48)(42,45)(46,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,22,31)(18,25,27,26)(20,23,32,24)(21,34,36,46)(28,43,35,47)(29,41,39,33)(30,38,45,37)(40,44,42,48);(17,20,24,19,22,32,23,31)(18,26,29,41,27,25,39,33)(21,34,38,30,36,46,37,45)(28,42,44,43,35,40,48,47);(2,7)(6,9)(8,15)(14,16)(18,27)(25,26)(28,35)(29,39)(33,41)(40,42)(43,47)(44,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21,24,38,22,36,23,37)(18,28,29,44,27,35,39,48)(19,30,32,46,31,45,20,34)(25,40,33,47,26,42,41,43)Polycyclic presentation:
2^9|b^2=f,c^2=h,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716872);