Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34655 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(11,12)(14,16)(17,18,21,25,20,24,23,31)(19,26,28,32,27,22,29,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15);(9,11,13,12)(10,14,15,16)(17,19,20,27)(18,22,24,26)(21,28,23,29)(25,30,31,32);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,18,23,31,20,24,21,25)(19,22,29,32,27,26,28,30);(1,3,4,7)(2,6,5,8)(19,27)(22,26)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=gh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6332876);