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 | |
2030 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)(5,6,9,8)(7,11,12,10)(13,14,17,16)(15,18,19,20)(21,22)(23,24)(25,33)(26,27)(28,29)(30,34)(31,35)(32,36);(5,6,9,8)(7,10,12,11)(15,19)(18,20)(22,27)(23,29)(25,34)(32,35);(5,7)(6,10)(8,11)(9,12)(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)(5,8,9,6)(7,11,12,10)(21,24,26,28)(22,29,27,23)(25,32,34,35)(30,36,33,31);(5,7)(6,10)(8,11)(9,12)(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=gh,b^2=fg,c^2=fh,d^2=g,e^2=fh,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, 1003567);