Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(21,22,26,28,25,27,23,24);(6,10)(11,12)(13,14,16,18)(15,19,20,17)(22,27)(24,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,17,18,19)(21,23,25,26)(22,24,27,28);(5,8,9,7)(6,11,10,12)(21,24,23,27,25,28,26,22)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=eg,d^2=eh,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239363, #72239393, #72239402, … |
| maximal normal subgroup of | #72239363, #72239393, #72239402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286686);