Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1682 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)(6,10)(7,12)(14,16)(18,20)(21,22,26,24)(23,29,30,27)(25,28,34,32)(31,36,33,35);(5,6)(7,11)(8,12)(9,10)(13,14,17,16)(15,18,19,20)(22,24)(23,30)(28,32)(31,33);(6,10)(8,11)(13,15)(14,18)(16,20)(17,19)(21,23,26,30)(22,27,24,29)(25,33,34,31)(28,36,32,35);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(21,24,26,22)(23,29,30,27)(25,32,34,28)(31,36,33,35);(5,8,9,11)(6,12,10,7)(13,16,17,14)(15,18,19,20)(21,25,26,34)(22,28,24,32)(23,31,30,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4507855);