Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395148 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10,14,15)(11,13,17,20)(12,19,22,16)(18,24,23,21);(1,2,4,3)(5,6)(7,8)(9,11,13,15,14,17,20,10)(12,18,21,19,22,23,24,16);(1,3,4,2)(6,8)(10,15)(11,17)(16,19)(18,23);(9,12,20,24,14,22,13,21)(10,16,17,23,15,19,11,18)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=fi,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308497);