Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,15,18)(11,13,17,14)(12,16,22,24)(19,23,20,21);(5,6)(7,8)(9,11,14,10,15,17,13,18)(12,20,21,24,22,19,23,16);(2,4)(6,8);(5,6)(7,8)(9,12)(10,16)(11,19)(13,23)(14,21)(15,22)(17,20)(18,24);(6,8)(9,13,15,14)(10,17,18,11)(12,21,22,23)(16,19,24,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=ehi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481529);