Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,21,18,22,16,19)(15,23,26,28,24,27,25,20);(5,7,8,12)(6,9,10,11)(14,19,22,21)(15,24)(20,23,28,27)(25,26);(2,4)(7,12)(9,11);(6,10)(9,11)(13,15,16,25,18,24,17,26)(14,20,21,23,22,28,19,27)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72317184, #72317285, #72317292 |
| maximal normal subgroup of | #72317184, #72317285, #72317292 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278373);