Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#61076 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
10386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,12,15)(9,17,19,33)(10,14,23,31)(11,24,27,16)(13,25,29,20)(18,35,36,30)(21,38,37,28)(22,34,26,32);(7,9)(8,13)(10,21)(11,25)(12,19)(14,30)(15,29)(16,33)(17,24)(18,22)(20,27)(23,37)(26,36)(28,32)(31,35)(34,38);(4,6);(7,10,12,23)(8,14,15,31)(9,18,19,36)(11,26,27,22)(13,28,29,38)(16,34,24,32)(17,35,33,30)(20,37,25,21);(8,15)(9,19)(14,31)(16,24)(18,36)(20,25)(21,37)(32,34)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483276);