Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535623 |
| 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 | |
1495 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)(3,7)(4,8)(5,6)(10,13)(11,15)(14,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32);(9,10,11,13,12,14,15,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(2,6)(7,8)(19,25)(22,28)(26,31)(29,32);(1,3,5,4)(2,7,6,8)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32);(1,4,5,3)(2,8,6,7)(20,27)(23,30)(26,31)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=g,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^d=eh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6472659);