Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1924 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)(11,15)(14,16)(17,18,21,24)(19,23,20,22)(25,26,28,30)(27,31,32,29);(1,2,5,3)(4,6,8,7)(9,10,13,12)(11,14,15,16)(17,19,21,20)(18,22,24,23)(27,32)(29,31);(2,3)(6,7)(17,18,21,24)(19,22,20,23)(25,26)(27,29)(28,30)(31,32);(1,3,5,2)(4,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,22)(21,24)(25,27)(26,29)(28,32)(30,31);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)(17,20,21,19)(18,23,24,22)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,c^2=fh,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2712072);