Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,20,21)(19,22,24,23);(1,2,4,3)(10,13)(14,16)(17,19)(18,22)(20,24)(21,23);(1,3,4,2)(9,11,12,15)(10,14,13,16)(18,21)(19,23)(22,24);(5,7,8,6)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=e,d^2=hi,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264283, #72264301, #72264365, … |
| maximal normal subgroup of | #72264283, #72264301, #72264365, … |
| maximal quotient of | #72321333 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266437);