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 | |
4686 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)(11,15)(12,20)(13,21)(14,19)(16,18)(17,23)(22,24);(2,4)(9,11,15,10)(12,22,23,19)(13,21,16,18)(14,17,24,20);(6,8)(9,12,15,23)(10,17,11,20)(13,24,16,14)(18,22,21,19);(9,13)(10,18)(11,21)(14,24)(15,16)(17,20);(9,14)(10,19)(11,22)(12,13)(15,24)(16,23)(17,18)(20,21)Polycyclic presentation:
2^9|b^2=i,c^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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, #72313664, #72315383, … |
| maximal normal subgroup of | #72313380, #72313664, #72315383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480943);