Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| 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 | #525 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1633 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)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,30)(20,35)(21,27)(22,26)(23,29)(24,32)(25,38)(28,41)(31,46)(33,44)(34,48)(36,42)(37,43)(39,45)(40,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,24)(20,36)(21,39)(22,32)(23,33)(25,27)(26,30)(28,44)(29,48)(31,38)(34,41)(35,47)(37,40)(42,43)(45,46);(17,20,22,37)(18,25,26,45)(19,31,32,21)(23,39,41,38)(24,42,30,47)(27,44,46,48)(28,43,29,35)(33,36,34,40);(2,7)(6,9)(8,15)(14,16)(18,26)(24,30)(25,45)(27,46)(28,29)(35,43)(42,47)(44,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21)(18,27)(19,20)(22,31)(23,36)(24,43)(25,44)(26,46)(28,47)(29,42)(30,35)(32,37)(33,38)(34,39)(40,41)(45,48)Polycyclic presentation:
2^9|b^2=fi,c^2=i,e^2=fi,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,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580785);