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×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,16,14,17,12,20)(11,15,18,23,19,24,21,22);(2,6)(4,7)(9,10)(11,15)(12,16)(13,20)(14,17)(18,22)(19,24)(21,23);(1,2)(3,4)(5,6)(7,8)(9,11,14,19)(10,15,17,24)(12,18,13,21)(16,22,20,23);(1,3,5,8)(2,7,6,4)(9,12,14,13)(10,16,17,20)(11,18,19,21)(15,22,24,23);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 6524810);