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 | |
1837 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)(23,31)(24,33)(25,34)(26,27)(28,32)(29,35)(30,36);(5,6,10,8)(7,11,9,12)(13,15)(14,16)(17,18)(19,20)(22,27)(23,32)(29,33)(30,34);(5,7,10,9)(6,11,8,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,28)(24,34)(25,29)(26,32)(27,31)(30,35)(33,36);(1,2,3,4)(5,8,10,6)(7,12,9,11)(21,24)(22,29)(23,30)(25,31)(26,35)(27,33)(28,36)(32,34);(5,9,10,7)(6,12,8,11)(21,25,26,36)(22,30,27,34)(23,29,32,33)(24,28,35,31)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fg,d^2=gi,e^2=g,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, 1316412);