Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,19)(12,21)(13,17)(14,18)(15,16)(20,23)(22,24);(3,7)(5,8)(9,11)(10,12)(13,20)(14,21)(15,22)(16,23)(17,19)(18,24);(1,2)(3,8)(4,6)(5,7)(9,12)(10,11)(13,19)(14,24)(15,23)(16,22)(17,20)(18,21);(10,16)(12,23)(17,18)(19,24);(9,13,15,14)(10,17,16,18)(11,20,22,21)(12,19,23,24)Polycyclic presentation:
2^9|e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258687);