Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25183 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,6,5)(7,8,13,19)(9,20,24,35)(10,26,11,16)(12,30,14,18)(15,25,32,23)(17,28,34,29)(21,33,37,36)(22,31,27,38);(3,5,6,4)(7,9,13,24)(8,15,19,32)(10,27,11,22)(12,25,14,23)(16,33,26,36)(17,31,34,38)(18,35,30,20)(21,29,37,28);(4,5)(7,10,13,11)(8,16,19,26)(9,21,24,37)(12,28,14,29)(15,31,32,38)(17,18,34,30)(20,36,35,33)(22,23,27,25);(7,11,13,10)(8,17,19,34)(9,22,24,27)(12,28,14,29)(15,31,32,38)(16,18,26,30)(20,33,35,36)(21,23,37,25)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346042);