Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,11,17,12,22,18,19,14,13,20,23,15,16,21,24);(5,6)(7,8)(9,11)(12,21)(14,20)(15,18)(16,22)(17,24)(19,23);(2,4)(6,8)(9,12,14,15)(10,16,13,22)(11,18,20,21)(17,24,23,19);(2,4)(9,13,14,10)(11,19,20,24)(12,22,15,16)(17,21,23,18)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72343483, #72343836, #72343842 |
| maximal normal subgroup of | #72343483, #72343836, #72343842 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416322);