Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,8,7)(9,10,13,17,12,16,22,15)(11,18,19,24,20,14,21,23);(1,2,4,3)(9,11,12,20)(10,14,16,18)(13,19,22,21)(15,24,17,23);(1,2,4,3)(10,15)(11,19)(13,22)(16,17)(20,21)(23,24);(1,3,4,2)(5,7,8,6)(10,16)(14,18)(15,17)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=ei,d^2=fhi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284849);