Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632008 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1559 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,22,30,21,25,20,24)(19,26,27,23,29,31,28,32);(1,3)(2,5)(4,7)(6,8)(9,10,12,13)(11,14,15,16);(3,7)(5,8)(10,13)(14,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,31)(25,32)(26,30);(3,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,24,25,30)(19,27,29,28)(23,31,32,26);(2,6)(5,8)(17,20,21,22)(18,24,25,30)(19,27,29,28)(23,31,32,26)Polycyclic presentation:
2^9|a^2=gi,b^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6542407);