Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(7,9)(11,12)(13,14,17,19)(15,16,18,20)(21,22,26,27)(23,30,29,25)(24,32,35,28)(31,36,33,34);(5,6,10,8)(7,11,9,12)(13,15)(14,16)(17,18)(19,20)(22,27)(23,29)(28,32)(31,33);(5,7,10,9)(6,11,8,12)(13,16)(14,18)(15,19)(17,20)(21,23,26,29)(22,25,27,30)(24,33,35,31)(28,34,32,36);(1,2,3,4)(5,8,10,6)(7,12,9,11)(21,24)(22,28)(23,31)(25,36)(26,35)(27,32)(29,33)(30,34);(5,9,10,7)(6,12,8,11)(21,25)(22,29)(23,27)(24,34)(26,30)(28,31)(32,33)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=g,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1316415);