Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,19,15,20,13,16)(11,18,22,17,21,24,12,23);(10,16,20,19)(11,21)(12,22)(17,24,23,18);(1,2,3,4)(9,11,14,22,15,21,13,12)(10,17,16,18,20,23,19,24);(6,8)(9,12,13,21,15,22,14,11)(10,18,19,17,20,24,16,23)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,c^b=ci,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72309721, #72309728, #72309747, … |
| maximal normal subgroup of | #72309721, #72309728, #72309747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300435);