Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33348 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C2×#534 | C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,15)(12,20,21,16)(13,17)(14,19,22,24)(18,23);(9,11)(10,14)(12,19)(13,18)(15,16)(17,22)(20,23)(21,24);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,16)(17,23)(20,22)(21,24);(11,18)(14,22)(15,23)(19,24)Polycyclic presentation:
2^9|a^2=f,c^2=h,f^2=h,b^a=be,e^a=eg,g^a=gi,d^b=di,f^b=fg,d^c=di,f^e=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387293);