Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #263707 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #24230 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,3)(4,5,8,7)(9,10)(11,19)(12,17)(13,16)(14,18)(15,21)(20,24)(22,23);(1,3,5,8,6,2,7,4)(9,11,12,21)(10,15,17,19)(13,22,14,20)(16,24,18,23);(2,3)(4,8)(10,16)(11,20)(13,14)(17,18)(21,22)(23,24);(1,4,5,3,6,8,7,2)(10,17)(15,19)(16,18)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,g^b=gi,d^c=dh,f^c=fi,g^c=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 170920);