Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,16)(11,15,20,21)(12,17,22,19)(14,18,23,24);(1,2,3,4)(9,11)(10,15)(12,18)(13,20)(14,19)(16,21)(17,23)(22,24);(6,8)(10,16)(11,19)(12,21)(14,23)(15,22)(17,20);(9,12)(10,17)(11,18)(13,22)(14,21)(15,23)(16,19)(20,24);(11,17)(12,21)(14,23)(15,22)(18,24)(19,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72263916, #72263937, #72263951, … |
| maximal normal subgroup of | #72263916, #72263937, #72263951, … |
| maximal quotient of | #72264795, #72265178, #72308258 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481366);