Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3888 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)(9,10,15,18,14,11,13,17)(12,16,22,24,21,19,20,23);(1,2,3,4)(9,11,14,10)(12,19,21,16)(13,17,15,18)(20,23,22,24);(9,12,14,21)(10,16,11,19)(13,22,15,20)(17,24,18,23);(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,e^b=ei,f^b=fi,h^b=hi,e^c=ei,h^c=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72302688, #72302839, #72302843 |
| maximal normal subgroup of | #72302688, #72302839, #72302843 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299847);