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 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,23,20)(19,26,29,32)(21,25,22,30)(24,31,27,28);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,23,29)(18,24,20,27)(21,28,22,31)(25,32,30,26);(17,18,23,20)(19,24,29,27)(21,25,22,30)(26,28,32,31);(17,20,23,18)(19,27,29,24)(21,25,22,30)(26,28,32,31);(17,21,23,22)(18,25,20,30)(19,28,29,31)(24,32,27,26)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,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, 6259301);