Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499712 |
| 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 | #13325 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
685 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)(7,10)(8,9)(11,12)(14,18)(19,26)(20,25)(24,28);(2,4)(14,19)(16,23)(17,22)(18,26)(20,28)(21,27)(24,25);(6,9)(10,12)(13,14,15,18)(16,24,21,28)(17,25,22,20)(19,27,26,23);(2,4)(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72343301, #72343826, #72343840 |
| maximal normal subgroup of | #72343301, #72343826, #72343840 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411589);