Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
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)(19,25)(20,21)(22,26)(23,27)(24,28)(29,33)(30,39)(31,43)(32,40)(34,35)(36,41)(37,44)(38,42)(45,47)(46,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,25)(20,33)(21,29)(22,36)(23,31)(24,32)(26,41)(27,43)(28,40)(30,35)(34,39)(37,46)(38,45)(42,47)(44,48);(17,20,21,18,24,27,23,28)(19,25,29,43,32,40,31,33)(22,34,35,26,38,44,37,42)(30,39,45,41,46,48,36,47);(2,7)(6,9)(8,15)(14,16)(17,21,24,23)(18,20,28,27)(19,29,32,31)(22,35,38,37)(25,33,40,43)(26,34,42,44)(30,45,46,36)(39,47,48,41);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22)(18,26)(19,30)(20,34)(21,35)(23,37)(24,38)(25,39)(27,44)(28,42)(29,45)(31,36)(32,46)(33,47)(40,48)(41,43)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=i,e^2=f,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=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593316);