Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352061 |
| 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 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,15,12)(11,18,21,24)(13,17,14,22)(16,23,19,20);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,19)(13,20)(14,23)(15,21)(17,24)(18,22);(1,3,5,4)(2,6,7,8)(9,10,15,12)(11,16,21,19)(13,17,14,22)(18,20,24,23);(9,12,15,10)(11,19,21,16)(13,17,14,22)(18,20,24,23);(9,13,15,14)(10,17,12,22)(11,20,21,23)(16,24,19,18)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259321);