Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| 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 | #60697 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,9)(3,10,12,7)(4,13,5,8)(11,15,14,16)(17,18,23,29)(19,31,35,25)(20,37,39,48)(21,30,24,27)(22,28,41,42)(26,34,46,38)(32,45,40,47)(33,43,36,44);(1,3,6,12)(2,7,9,10)(4,11,5,14)(8,15,13,16)(17,19)(18,25)(20,22)(21,33)(23,35)(24,36)(26,45)(27,43)(28,48)(29,31)(30,44)(32,34)(37,42)(38,40)(39,41)(46,47);(17,20,21,38,23,39,24,34)(18,26,30,48,29,46,27,37)(19,32,36,41,35,40,33,22)(25,42,43,47,31,28,44,45);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,23,24)(18,27,29,30)(19,33,35,36)(20,38,39,34)(22,40,41,32)(25,43,31,44)(26,37,46,48)(28,45,42,47);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,22,24,40,23,41,21,32)(18,28,27,47,29,42,30,45)(19,34,33,39,35,38,36,20)(25,26,44,37,31,46,43,48)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=hi,d^2=f,e^2=fhi,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,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719194);