Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4648 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)(5,6)(7,8)(9,10,13,15)(11,12,16,22)(14,17,23,24)(18,21,19,20);(2,4)(9,11,13,16)(10,12,15,22)(14,18,23,19)(17,21,24,20);(6,8)(10,15)(11,18)(12,20)(14,23)(16,19)(21,22);(9,12)(10,16)(11,15)(13,22)(14,21)(17,19)(18,24)(20,23);(11,18)(12,21)(14,23)(16,19)(17,24)(20,22)Polycyclic presentation:
2^9|a^2=i,b^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313380, #72313685, #72315383, … |
| maximal normal subgroup of | #72313380, #72313685, #72315383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480655);