Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,22,28,23,29,39,27)(19,30,34,24,33,42,48,41)(20,35,38,44,21,26,37,25)(31,43,32,46,36,40,47,45);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,22,34,23,33,39,48)(18,24,27,30,29,41,28,42)(20,36,37,32,21,31,38,47)(25,43,35,46,44,40,26,45);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,25)(19,31)(21,23)(22,37)(24,40)(26,28)(27,35)(29,44)(30,45)(32,48)(33,36)(34,47)(38,39)(41,43)(42,46);(3,10)(6,9)(11,16)(14,15)(17,21,23,20)(18,26,29,35)(19,32,33,47)(22,37,39,38)(24,40,41,43)(25,27,44,28)(30,46,42,45)(31,34,36,48);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(19,33)(24,41)(30,42)(31,36)(32,47)(34,48)(40,43)(45,46)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7090016);