Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,8)(5,7)(10,14,16,20)(15,17,22,23);(1,3,5,4)(2,6,7,8)(9,10)(11,17)(12,20)(13,16)(14,21)(15,19)(18,23)(22,24);(1,4,5,3)(2,6,7,8);(9,11,12,19,13,18,21,24)(10,15,14,17,16,22,20,23);(11,18)(15,22)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=f,f^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287823);