Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#39519 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,7,6)(5,8,10,9)(12,15)(16,18)(19,20)(21,27)(22,26)(23,24)(25,29)(28,33)(30,31)(32,34);(3,5)(4,8)(6,9)(7,10)(20,24)(25,27)(26,31)(32,33);(3,4,7,6)(5,9,10,8)(11,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,26)(23,24)(27,29)(28,32)(30,31)(33,34);(11,13,14,17)(12,16,15,18)(19,21)(20,25)(22,28)(23,29)(24,27)(26,32)(30,34)(31,33);(3,6,7,4)(5,8,10,9)(19,22,23,30)(20,26,24,31)(21,28,29,34)(25,32,27,33)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7543968);