Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(6,7)(9,10,13,14)(11,15,19,17)(12,16,21,22)(18,23,24,20);(5,6,8,7)(9,11)(10,15)(12,20)(13,19)(14,17)(16,18)(21,23)(22,24);(2,4)(5,7,8,6)(10,14)(11,17)(15,19)(16,22)(18,20)(23,24);(9,12,13,21)(10,16,14,22)(11,18,19,24)(15,23,17,20);(11,19)(15,17)(18,24)(20,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345865);