Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397902 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | #34936 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,18)(12,23)(13,16)(14,17)(15,20)(19,21)(22,24);(2,5)(6,7)(9,11,12,20,14,21,13,22)(10,15,16,18,17,24,23,19);(9,10)(11,19,21,18)(12,23)(13,16)(14,17)(15,20,24,22);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|b^2=eh,c^2=f,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72237198, #72239447, #72252487, … |
| maximal normal subgroup of | #72237198, #72239447, #72252487, … |
| maximal quotient of | #72237249, #72257355, #72257548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323812);