Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(6,7)(9,10)(11,19)(12,20)(13,16)(14,18)(15,21)(17,23)(22,24);(5,6,8,7)(9,11,14,15,13,20,23,24)(10,12,18,22,16,19,17,21);(5,7,8,6)(9,12,13,19)(10,15,16,24)(11,18,20,17)(14,22,23,21);(10,16)(11,20)(15,24)(17,18)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=di,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72238812, #72238843, #72239940 |
| maximal normal subgroup of | #72238812, #72238843, #72239940 |
| maximal quotient of | #72238820, #72238846 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260709);