Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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 | C22×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10,14,18,15,19,13,23)(11,20,21,24,22,17,12,16);(1,2)(3,4)(5,7,8,6)(9,11,13,12,15,22,14,21)(10,16,18,20,19,24,23,17);(2,4)(6,7);(9,12,13,22,15,21,14,11)(10,17,18,16,19,20,23,24)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321891, #72321898, #72321910, … |
| maximal normal subgroup of | #72321891, #72321898, #72321910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267676);