Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509065 |
| 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 | #28227 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
635 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)(7,8)(9,11)(10,12)(13,14,16,18)(15,21,23,19)(17,20,24,25)(22,27,28,26);(5,7)(6,10)(8,11)(9,12)(14,18)(15,22)(17,25)(19,27)(20,24)(21,26)(23,28);(2,4)(5,8,9,10)(6,7,11,12)(13,15)(14,19)(16,23)(17,26)(18,21)(20,28)(22,25)(24,27);(2,4)(6,11)(7,12)(13,14,16,18)(15,19,23,21)(17,20,24,25)(22,26,28,27)Polycyclic presentation:
2^9|a^2=i,c^2=ei,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416461);