Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2406 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)(9,10,12,14)(11,15,16,13)(18,22)(19,20)(23,24);(5,6)(7,8)(9,11,12,16)(10,13,14,15)(17,18,21,22)(19,23,24,20);(2,4)(6,8)(17,19,21,24)(18,23,22,20);(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=ei,b^2=e,c^2=i,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 | #72311814, #72311827, #72311835, … |
| maximal normal subgroup of | #72311814, #72311827, #72311835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301649);