Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13304 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(14,20)(15,19)(16,24)(17,25)(21,27)(22,26)(23,28);(1,2)(3,4)(13,14)(15,21)(16,18)(17,26)(19,22)(20,25)(23,24)(27,28);(2,4)(6,9)(10,12)(13,15,18,19)(14,21,16,22)(17,23,28,25)(20,26,24,27);(5,7,8,11)(6,10,9,12)(13,16,18,14)(15,22,19,21)(17,27,28,26)(20,25,24,23)Polycyclic presentation:
2^9|c^2=i,d^2=f,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=eh,h^b=hi,e^d=eh,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318893, #72319560, #72319591, … |
| maximal normal subgroup of | #72318893, #72319560, #72319591, … |
| maximal quotient of | #72322616 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411558);