Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| 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 | |
1209 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,15,18,13,12,14,17)(11,19,22,23,20,16,21,24);(1,3,5,4)(2,6,8,7)(9,11,15,22,13,20,14,21)(10,16,18,24,12,19,17,23);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,19,20,16)(14,17,15,18)(21,23,22,24);(1,4,5,3)(2,6,8,7)(9,10,13,12)(11,16,20,19)(14,17,15,18)(21,23,22,24)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=e,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305068);