Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1840 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)(4,7)(6,8)(9,10,13,14)(11,12,15,16)(19,23)(22,24)(25,26,29,28)(27,31,32,30);(1,3,6,4)(2,7,8,5)(10,14)(11,15)(25,27,29,32)(26,30,28,31);(2,8)(3,4)(9,11,13,15)(10,12,14,16)(17,18,21,20)(19,22,23,24)(25,26,29,28)(27,30,32,31);(1,4,6,3)(2,7,8,5)(9,11)(10,12)(13,15)(14,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32);(1,5,6,7)(2,4,8,3)(9,12)(10,15)(11,14)(13,16)(17,20,21,18)(19,24,23,22)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fgi,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,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, 2665754);