Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#546 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,4)(6,8)(9,10,12,13)(11,15,16,14)(17,18,21,22,20,23,19,24);(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16);(1,3,5,4)(2,6,7,8)(10,13)(14,15)(17,19,20,21)(18,22,23,24);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,18,19,24,20,23,21,22);(9,11,12,16)(10,14,13,15)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=gh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6332982);