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 | |
3212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,15)(11,18)(12,19)(13,23)(16,17)(20,21);(1,2,4,5)(3,6,7,8)(9,10,14,16)(11,19,21,12)(13,17,23,15)(18,24,20,22);(2,5)(3,7)(9,11,14,21)(10,12,16,19)(13,20,23,18)(15,22,17,24);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,22)(14,19)(15,20)(16,21)(17,18)(23,24);(2,5)(3,7)(10,16)(12,19)(15,17)(22,24)Polycyclic presentation:
2^9|b^2=h,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503056);