Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1563 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)(3,6)(4,8)(5,7)(17,18,21,24,22,25,20,29)(19,26,27,32,28,31,30,23);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26);(2,7)(6,8)(10,13)(11,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26);(1,4,5,3)(2,8,7,6)(9,11)(10,14)(12,15)(13,16)(18,25)(23,32)(24,29)(26,31);(10,13)(11,15)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=i,d^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6520162);