Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470058 |
| 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 | |
3373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10)(11,19)(12,18)(13,16)(14,17)(15,20)(21,24)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,17,23)(12,22,13,20)(16,24,18,19);(2,5)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,23)(18,24);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,19,23,24);(1,3,4,7)(2,6,5,8)(10,17)(15,23)(16,18)(19,24)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398169);