Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | |
1673 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)(19,31)(20,35)(21,30)(22,28)(23,29)(24,27)(25,33)(26,37)(32,43)(34,45)(36,47)(38,44)(39,46)(40,42)(41,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,33)(18,25,29,31)(20,36,38,40)(21,32,24,34)(22,37,41,39)(26,28,46,48)(27,43,30,45)(35,42,44,47);(17,20,23,38)(18,26,29,46)(19,22,33,41)(21,37,24,39)(25,42,31,47)(27,35,30,44)(28,43,48,45)(32,36,34,40);(2,7)(6,9)(8,15)(14,16)(17,21,23,24)(18,27,29,30)(19,32,33,34)(20,37,38,39)(22,36,41,40)(25,43,31,45)(26,35,46,44)(28,42,48,47);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,23,41)(18,28,29,48)(19,20,33,38)(21,40,24,36)(25,44,31,35)(26,43,46,45)(27,47,30,42)(32,39,34,37)Polycyclic presentation:
2^9|b^2=f,c^2=i,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,g^c=gi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580998);