Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470028 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,22)(18,24)(21,23);(3,4)(6,7)(9,11)(10,14)(12,21)(13,18)(15,23)(16,24)(17,20)(19,22);(1,3,5,4)(2,6,8,7)(11,18)(12,22)(14,23)(16,20);(1,4,5,3)(2,7,8,6)(10,15)(11,18)(16,20)(19,21);(2,8)(6,7)(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|c^2=hi,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257626, #72257636, #72257704, … |
| maximal normal subgroup of | #72257626, #72257636, #72257704, … |
| maximal quotient of | #72257956, #72264431, #72264462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394694);