Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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 | |
2173 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)(9,10,15,14)(11,20,23,17)(12,24,22,13)(16,19,21,18);(2,5)(6,8)(9,11)(10,17)(12,19)(13,21)(14,20)(15,23)(16,24)(18,22);(9,12)(10,11)(13,18)(14,23)(15,22)(16,20)(17,21)(19,24);(1,3,4,7)(2,6,5,8)(9,13,15,24)(10,12,14,22)(11,21,23,16)(17,19,20,18);(9,14)(10,15)(11,22)(12,23)(13,17)(16,19)(18,21)(20,24)Polycyclic presentation:
2^9|a^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482644);