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 |
| prime radical | #61179 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1693 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,30)(20,33)(21,38)(22,28)(23,29)(24,27)(25,32)(26,37)(31,43)(34,47)(35,46)(36,44)(39,42)(40,48)(41,45);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,32)(18,25,29,30)(20,34,36,39)(21,31,24,41)(22,37,40,35)(26,28,46,48)(27,43,38,45)(33,42,44,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,26)(19,22)(21,35)(23,36)(24,37)(25,42)(27,44)(28,43)(29,46)(30,47)(31,39)(32,40)(33,38)(34,41)(45,48);(2,8)(6,9)(7,14)(13,15)(17,21,23,24)(18,27,29,38)(19,31,32,41)(20,35,36,37)(22,39,40,34)(25,43,30,45)(26,44,46,33)(28,47,48,42);(3,11)(6,9)(10,16)(13,15)(17,22)(18,28)(19,20)(21,34)(23,40)(24,39)(25,44)(26,43)(27,42)(29,48)(30,33)(31,37)(32,36)(35,41)(38,47)(45,46)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=i,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, 6577110);