Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10,14,20)(11,19,15,16)(12,22,13,18)(17,23,24,21);(1,3,4,2)(6,7)(9,11,14,15)(10,16,20,19)(12,21,13,23)(17,22,24,18);(9,12,14,13)(10,17,20,24)(11,21,15,23)(16,22,19,18);(5,7,8,6)(9,13,14,12)(10,18,20,22)(11,21,15,23)(16,24,19,17);(9,11,14,15)(10,19,20,16)(12,21,13,23)(17,18,24,22)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,g^c=gi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289080);