Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573671 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,15,19,14,11,13,17)(12,18,22,23,20,16,21,24);(1,2,4,5)(3,6,7,8)(9,11,14,10)(12,18,20,16)(13,17,15,19)(21,23,22,24);(2,5)(3,7)(10,11)(12,20)(13,15)(23,24);(1,3)(2,6)(4,7)(5,8)(9,12,14,20)(10,16,11,18)(13,21,15,22)(17,23,19,24);(2,5)(3,7)(10,11)(12,20)(17,19)(21,22)Polycyclic presentation:
2^9|a^2=fi,b^2=h,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503373);