Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,14,17,15,11,13,19)(12,16,21,23,22,18,20,24);(5,6,7,8)(9,11,15,10)(12,16,22,18)(13,17,14,19)(20,24,21,23);(2,4);(9,12,14,21,15,22,13,20)(10,16,17,23,11,18,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,d^b=di,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72309702, D4×#1765, #72310137, … |
| maximal normal subgroup of | #72309702, D4×#1765, #72310137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261504);