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 | |
1635 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)(19,25)(20,34)(21,27)(22,39)(23,28)(24,29)(26,36)(30,48)(31,42)(32,44)(33,45)(35,41)(37,46)(38,47)(40,43);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19)(18,25)(20,35)(21,36)(22,32)(23,40)(24,33)(26,27)(28,43)(29,45)(30,37)(31,38)(34,41)(39,44)(42,47)(46,48);(17,20,24,31)(18,26,29,46)(19,30,33,21)(22,37,23,36)(25,41,45,47)(27,44,48,43)(28,42,39,34)(32,38,40,35);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15);(3,11)(6,9)(10,16)(14,15)(17,21,24,30)(18,27,29,48)(19,31,33,20)(22,38,23,35)(25,42,45,34)(26,44,46,43)(28,41,39,47)(32,37,40,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=f,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, 6579485);