Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2122 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)(4,5)(7,8)(9,10)(11,17)(12,22)(13,18)(14,20)(15,19)(16,23)(21,24);(2,5)(6,8)(9,11)(10,12)(13,22)(14,16)(15,24)(17,18)(19,23)(20,21);(9,12,13,17)(10,16,18,24)(11,20,22,19)(14,21,15,23);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,18)(13,22)(14,16)(15,24)(19,21)(20,23);(9,10)(11,21)(12,16)(13,18)(14,19)(15,20)(17,24)(22,23)Polycyclic presentation:
2^9|c^2=i,d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482980);