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 | #3012 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1628 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)(17,18,21,26)(19,29,32,24)(20,27,35,42)(22,25,23,28)(30,40,33,39)(31,45,47,38)(34,48,46,43)(36,41,37,44);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19)(18,24)(20,34)(21,32)(22,33)(23,30)(25,40)(26,29)(27,43)(28,39)(31,37)(35,46)(36,47)(38,44)(41,45)(42,48);(18,25)(19,30)(22,23)(26,28)(27,41)(31,46)(32,33)(34,47)(36,37)(39,40)(42,44)(43,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,26)(20,35)(24,29)(25,28)(31,47)(34,46)(36,37)(39,40);(3,11)(6,9)(10,16)(14,15)(17,20)(18,27)(19,31)(21,35)(22,36)(23,37)(24,38)(25,41)(26,42)(28,44)(29,45)(30,46)(32,47)(33,34)(39,48)(40,43)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^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,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, 6579834);