Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500002 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,14)(12,20,23,24)(13,19)(15,17)(16,22,18,21);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,16,23,18)(13,19,15,17)(20,22,24,21);(2,5)(3,7)(9,12)(10,16)(11,18)(13,22)(14,23)(15,21)(17,24)(19,20);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,17,11,19)(12,21,23,22)(16,20,18,24);(2,5)(6,8)(10,11)(16,18)(17,19)(20,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420199);