Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993915 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(11,15)(14,16)(17,18,20,25,21,23,22,26)(19,27,30,24,29,32,28,31);(1,3)(2,6)(4,7)(5,8)(10,13)(11,15)(18,23)(24,31)(25,26)(27,32);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,29)(18,24,23,31)(20,28,22,30)(25,27,26,32);(3,8)(6,7)(10,13)(14,16)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,27,31,32);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=fg,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72283907, #72283935, #72284125, … |
| maximal normal subgroup of | #72283907, #72283935, #72284125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7057251);