Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31076 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#32702 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | #2782 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,12,9,14,7,13)(5,15,18,10,19,26,17,27)(6,20,23,11,24,29,22,30)(16,31,32,28,33,25,21,34);(3,5)(4,10)(6,21)(7,17)(8,18)(9,19)(11,28)(12,26)(13,15)(14,27)(16,23)(20,31)(22,33)(24,32)(25,29)(30,34);(3,6,7,22,9,24,8,23)(4,11,12,29,14,30,13,20)(5,16,18,32,19,33,17,21)(10,25,15,28,27,31,26,34);(5,17,19,18)(10,26,27,15)(16,21,33,32)(25,34,31,28)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421147);