Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18,22,25)(19,23,21,29)(20,27,28,31)(24,30,32,26);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(10,13)(14,16)(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27);(1,3,4,7)(2,6,5,8)(11,15)(14,16)(18,25)(23,29)(24,32)(27,31);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6442360);