Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574070 |
| 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 | |
1850 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)(9,10,15,14)(11,18,22,17)(12,23,13,19)(16,21,24,20);(2,6)(7,8)(9,11)(10,17)(12,24)(13,16)(14,18)(15,22)(19,20)(21,23);(9,12,16,11,15,13,24,22)(10,18,20,23,14,17,21,19);(1,3,5,4)(2,7,6,8)(9,13,15,12)(10,19,14,23)(11,16,22,24)(17,20,18,21);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,19,22,23)(12,17,13,18)(16,21,24,20)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492248);