Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574100 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| 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 | |
1725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,12,15,13,17,19,16)(11,18,21,14,20,23,24,22);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,22)(23,24);(2,5)(3,7)(9,11,19,24,13,20,12,21)(10,14,15,23,17,22,16,18);(1,3)(2,6)(4,7)(5,8)(11,20)(14,22)(18,23)(21,24);(2,5)(3,7)(11,20)(14,22)(18,23)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=gi,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513851);