Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1762 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)(5,6,10,11)(7,9,8,12)(13,14)(15,18)(16,17)(19,20)(22,24)(23,29)(28,32)(31,33);(5,7,10,8)(6,9,11,12)(14,17)(15,19)(21,22,26,24)(23,30,29,27)(25,28,34,32)(31,36,33,35);(6,11)(7,8)(13,15,16,19)(14,18,17,20)(21,23,26,29)(22,27,24,30)(25,33,34,31)(28,36,32,35);(1,2,3,4)(5,8,10,7)(6,9,11,12)(21,24,26,22)(23,27,29,30)(25,32,34,28)(31,35,33,36);(5,9)(6,7)(8,11)(10,12)(13,14)(15,20)(16,17)(18,19)(21,25,26,34)(22,28,24,32)(23,31,29,33)(27,35,30,36)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=f,d^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5417192);