Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417566 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,18,19,24)(12,15,21,22)(13,16)(14,17)(20,23);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,22)(14,19)(15,23)(16,24)(17,21)(18,20);(2,7)(6,8)(11,19)(13,23)(16,20)(18,24);(1,4,5,3)(2,8,7,6)(9,12,14,21)(10,15,17,22)(11,20,19,16)(13,24,23,18)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,d^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359367);