Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(2,6)(7,8)(12,20)(15,22)(18,24)(21,23);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23);(1,3,5,4)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=ci,f^a=fh,g^b=gh,d^c=dh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249766);