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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2524 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18,21,23)(19,26,27,32)(20,25,22,29)(24,30,31,28);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(19,27)(25,29)(28,30);(11,15)(13,16)(17,19,21,27)(18,24,23,31)(20,28,22,30)(25,32,29,26);(1,4,5,3)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(19,27)(24,31)(26,32)(28,30);(11,15)(13,16)(17,20,21,22)(18,25,23,29)(19,28,27,30)(24,32,31,26)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=ehi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6500111);