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 | |
1641 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)(19,24)(20,32)(21,36)(22,26)(23,27)(25,35)(28,42)(29,37)(30,40)(31,41)(33,47)(34,44)(38,43)(39,46)(45,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,31)(18,24,27,41)(20,33,34,46)(21,30,22,37)(25,42,43,48)(26,29,36,40)(28,38,45,35)(32,47,44,39);(17,20,21,35,23,34,22,38)(18,25,26,32,27,43,36,44)(19,28,30,46,31,45,37,33)(24,39,29,42,41,47,40,48);(2,7)(6,9)(8,15)(14,16)(20,34)(25,43)(28,45)(32,44)(33,46)(35,38)(39,47)(42,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,18)(19,29)(20,25)(21,26)(22,36)(23,27)(24,37)(28,42)(30,41)(31,40)(32,35)(33,39)(34,43)(38,44)(45,48)(46,47)Polycyclic presentation:
2^9|b^2=fi,c^2=g,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, 6593309);