Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,15)(12,16)(13,14)(18,22)(19,24)(25,26,28,30)(27,31,32,29);(1,3)(2,7)(4,8)(5,6)(9,10,13,14)(11,15,16,12)(17,18,21,22)(19,23,24,20)(27,32)(29,31);(3,5)(7,8)(10,14)(12,15)(17,19)(18,23)(20,22)(21,24)(25,26)(27,29)(28,30)(31,32);(1,4,6,2)(3,7,5,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,22)(21,24)(25,27)(26,29)(28,32)(30,31);(1,5,6,3)(2,8,4,7)(9,12,13,15)(10,16,14,11)(17,20,21,23)(18,24,22,19)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fghi,d^2=ghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3875004);