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 | |
1924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,18)(14,21)(15,19)(16,20)(17,22)(23,24);(2,6)(7,8)(9,11)(10,12)(13,16)(14,23)(15,17)(18,20)(19,22)(21,24);(9,12)(10,17)(11,21)(13,19)(14,24)(15,22)(16,18)(20,23);(1,3,5,4)(2,7,6,8)(9,13,16,11)(10,18,20,12)(14,17,15,23)(19,24,21,22);(1,4,5,3)(2,8,6,7)(9,10)(11,22)(12,17)(13,24)(14,19)(15,21)(16,20)(18,23)Polycyclic presentation:
2^9|d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,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, 6485232);