Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1617 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(10,14)(11,15)(17,18)(19,20)(21,24)(22,23)(27,32)(29,31);(2,3)(6,7)(9,10,13,14)(11,16,15,12)(17,19,21,23)(18,22,24,20)(25,26,30,28)(27,29,32,31);(4,8)(6,7)(10,14)(12,16)(17,20,21,22)(18,23,24,19)(25,27,30,32)(26,31,28,29);(1,3,5,2)(4,6,8,7)(9,11,13,15)(10,12,14,16)(17,19)(18,20)(21,23)(22,24)(25,28,30,26)(27,29,32,31);(1,4,5,8)(2,6,3,7)(9,12)(10,11)(13,16)(14,15)(17,20)(18,23)(19,24)(21,22)(25,29)(26,27)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=g,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3966998);