Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#3195 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
8798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2);(3,4);(5,6)(7,11)(8,10)(9,12)(13,14,17,20)(15,21,22,28)(16,24,26,19)(18,25,27,23);(5,7)(6,9)(8,12)(10,11)(15,22)(18,27)(21,28)(23,25);(7,12)(9,11)(13,15,17,22)(14,18,20,27)(16,25,26,23)(19,28,24,21);(6,10)(9,11)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482596);