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 | #60572 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1621 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18,22,27)(19,30,31,25)(20,33,35,43)(21,28,38,47)(23,39,24,29)(26,37,46,36)(32,44,40,45)(34,48,41,42);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19,22,31)(18,25,27,30)(20,34,35,41)(21,36,38,37)(23,32,24,40)(26,28,46,47)(29,44,39,45)(33,42,43,48);(17,20,22,35)(18,26,27,46)(19,21,31,38)(23,37,24,36)(25,42,30,48)(28,45,47,44)(29,43,39,33)(32,41,40,34);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,27)(25,30)(26,46)(28,47)(29,39)(33,43)(42,48)(44,45);(3,11)(6,9)(10,16)(14,15)(17,21,22,38)(18,28,27,47)(19,20,31,35)(23,34,24,41)(25,43,30,33)(26,45,46,44)(29,42,39,48)(32,36,40,37)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=fi,e^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,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579491);