Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500006 |
| 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 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,15,16,22)(11,18)(17,24,19,23)(20,21);(1,2)(3,6)(4,5)(7,8)(10,16)(12,13)(17,19)(20,21);(2,5)(3,7)(9,10)(11,19)(12,15)(13,22)(14,16)(17,18)(20,23)(21,24);(1,3,4,7)(2,6,5,8)(9,11,14,18)(10,17,16,19)(12,21,13,20)(15,23,22,24);(2,5)(6,8)(9,12,14,13)(10,15,16,22)(11,20,18,21)(17,24,19,23)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420517);