Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×#1333 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#102 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4);(5,6,7,9,8,10,11,12)(13,14,16,20,17,22,27,21)(15,23,25,19,26,18,24,28);(14,18)(15,24)(16,27)(19,20)(21,28)(22,23)(25,26)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216549, #72216590, C4×#12067, … |
| maximal normal subgroup of | #72216549, #72216590, C4×#12067, … |
| maximal quotient of | #72216529, C8×#1333, C2×C4×#1333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35731);