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 | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,7)(9,10,13,17,14,11,18,16)(12,15,20,23,21,19,24,22);(1,2)(3,4)(5,6)(7,8);(2,4)(5,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,23)(22,24);(9,12,13,20,14,21,18,24)(10,15,17,23,11,19,16,22)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^c=di,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72216198, #72309702, #72310083, … |
| maximal normal subgroup of | #72216198, #72309702, #72310083, … |
| maximal quotient of | #72215970 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266346);