Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61559 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C2×#3483 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10)(11,20)(12,17)(13,18)(14,23)(15,19)(16,21)(22,24);(1,3,6,2)(4,8,5,7)(9,11,15,12)(10,16,19,24)(13,21,14,22)(17,23,20,18);(1,4,6,5)(2,7,3,8)(9,12,15,11)(10,17,19,20)(13,21,14,22)(16,23,24,18);(1,4,6,5)(2,7,3,8)(9,13,15,14)(10,18,19,23)(11,21,12,22)(16,17,24,20);(10,19)(16,24)(17,20)(18,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,e^a=ehi,f^a=fh,f^b=fh,d^c=dhi122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250333);