Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1414 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)(6,7)(9,10)(11,16)(12,15)(13,14)(17,18,21,24,23,19,22,20);(2,4)(5,6,8,7)(10,14)(15,16)(17,19,23,18)(20,21,24,22);(5,6,8,7)(9,11,13,12)(10,15,14,16)(17,20,22,19,23,24,21,18);(5,7,8,6)(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,20,19,24)Polycyclic presentation:
2^9|a^2=efi,b^2=h,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237235, #72238785, #72239610, … |
| maximal normal subgroup of | #72237235, #72238785, #72239610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317861);