Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 | |
4631 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)(11,16)(12,21)(13,18)(14,17)(15,19)(20,22)(23,24);(2,4)(11,18)(12,14)(13,16)(17,21)(20,24)(22,23);(6,8)(9,11)(10,16)(12,18)(13,21)(14,24)(15,20)(17,23)(19,22);(9,12)(10,17)(11,20)(13,23)(14,15)(19,21);(9,13,15,23)(10,18,19,24)(11,21,20,17)(12,22,14,16)Polycyclic presentation:
2^9|e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72313380, #72313623, #72315401, … |
| maximal normal subgroup of | #72313380, #72313623, #72315401, … |
| maximal quotient of | #72315807 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480723);