Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,13,15)(11,16,12,14)(17,18,19,22)(20,23,21,24);(5,6)(7,8)(9,11,13,12)(10,14,15,16)(18,22)(23,24);(2,4)(6,8)(9,12,13,11)(10,14,15,16)(18,23)(20,21)(22,24);(18,24)(20,21)(22,23)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264161, #72264202, #72264224, … |
| maximal normal subgroup of | #72264161, #72264202, #72264224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302087);