Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,23,24,20)(25,26)(27,31)(28,29)(30,32);(2,5)(6,8)(9,10,12,14)(11,13,16,15)(17,19,21,24)(18,20,22,23)(26,29)(27,32);(3,7)(6,8)(11,16)(13,15)(18,22)(20,23)(25,26,28,29)(27,30,32,31);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,20)(18,19)(21,23)(22,24)(25,26)(27,30)(28,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,27,28,32)(26,30,29,31)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1623433);