Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,10)(7,11,12,8)(15,19)(17,20)(21,22)(23,27)(24,26)(25,28)(29,30)(31,35)(32,34)(33,36);(5,6)(7,11)(8,12)(9,10)(13,14,16,18)(15,17,19,20)(22,24)(23,29)(28,32)(31,33);(6,10)(8,11)(13,14)(15,17)(16,18)(19,20)(21,23,26,29)(22,27,24,30)(25,33,34,31)(28,36,32,35);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,15,16,19)(14,17,18,20)(21,24,26,22)(23,30,29,27)(25,32,34,28)(31,36,33,35);(5,8,9,11)(6,12,10,7)(21,25,26,34)(22,28,24,32)(23,31,29,33)(27,35,30,36)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1993615);