Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C42 |
| direct product | C2×#14657 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C23×#120 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
13798 conjugacy classes of subgroups; only the 1346 normal subgroups are shown.
This group has 13798 conjugacy classes of subgroups — too many to draw. Only its 1346 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(12,15)(16,18)(19,20)(21,24)(22,23)(25,26);(4,6)(7,8,9,10)(12,15)(13,17)(20,23)(21,25);(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,26)(22,23)(24,25);(11,13)(12,16)(14,17)(15,18)(19,21)(20,24)(22,25)(23,26)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#5021138, #72210046, #72210050, … |
| maximal normal subgroup of | C2×#5021138, #72210046, #72210050, … |
| maximal quotient of | C2×#391709, C2×#391710, #72258272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 451558);