Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | |
1954 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)(13,14,17,16)(15,18,19,20)(21,22)(23,24)(25,33)(26,27)(28,29)(30,34)(31,35)(32,36);(5,6,10,8)(7,11,9,12)(15,19)(18,20)(22,27)(23,29)(25,34)(32,35);(5,7,10,9)(6,11,8,12)(13,15,17,19)(14,18,16,20)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,33)(34,36);(1,2)(3,4)(5,8,10,6)(7,12,9,11)(21,24,26,28)(22,29,27,23)(25,32,34,35)(30,36,33,31);(5,9,10,7)(6,12,8,11)(13,16,17,14)(15,20,19,18)(21,25)(22,30)(23,31)(24,32)(26,34)(27,33)(28,35)(29,36)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=gh,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1004156);