Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,16)(12,21)(13,18)(14,19)(15,20)(17,24)(22,23);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,18)(13,21)(14,23)(15,17)(19,22)(20,24);(9,12)(10,17)(11,20)(13,24)(14,16)(15,22)(18,23)(19,21);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,18,20,19)(11,21,17,23)(12,24,22,16);(1,4,5,3)(2,6,7,8)(9,10)(11,22)(12,17)(13,19)(14,18)(15,20)(16,23)(21,24)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6583714);