Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10,14,18,12,16,13,17)(11,19,20,23,21,15,22,24);(1,2,4,3)(5,6,7,8)(9,11,14,20,12,21,13,22)(10,15,18,24,16,19,17,23);(6,8)(10,16)(11,20)(13,14)(15,23)(19,24)(21,22);(1,3,4,2)(10,16)(11,21)(17,18)(20,22)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216155, #72238763, #72239142, … |
| maximal normal subgroup of | #72216155, #72238763, #72239142, … |
| maximal quotient of | #72216082 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269205);