Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533075 |
| 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×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,19)(12,21)(13,16)(14,18)(15,24)(17,20)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,13)(14,20)(15,21)(16,22)(18,23)(19,24);(2,5)(6,8)(9,12)(10,11)(13,22)(14,19)(15,23)(16,17)(18,20)(21,24);(1,3)(2,6)(4,7)(5,8)(9,13,16,10)(11,12,22,17)(14,18,15,24)(19,20,23,21);(3,7)(6,8)(9,14,16,15)(10,18,13,24)(11,20,22,21)(12,19,17,23)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6440466);