Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2862 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)(6,8)(9,10)(11,19)(12,16)(13,24)(14,18)(15,17)(20,23)(21,22);(2,4)(9,10)(11,20)(12,22)(13,17)(14,18)(15,24)(16,21)(19,23);(5,6,7,8)(9,11)(10,16)(12,14)(13,21)(15,23)(17,22)(18,19)(20,24);(9,12)(10,16)(11,14)(13,23)(15,21)(17,22)(18,19)(20,24);(9,13,14,15)(10,17,18,24)(11,21,12,23)(16,22,19,20)Polycyclic presentation:
2^9|a^2=f,c^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350026);