Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397950 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3151 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1952 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)(9,10,15,18)(11,19,22,20)(12,13,17,14)(16,23,24,21);(2,5)(6,7)(9,11,13,21,15,22,14,23)(10,16,12,19,18,24,17,20);(9,12,14,10,15,17,13,18)(11,20,21,16,22,19,23,24);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=i,b^2=eh,c^2=f,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72227229, #72227262, #72227274, … |
| maximal normal subgroup of | #72227229, #72227262, #72227274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323814);