Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1758 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)(5,6,9,10)(7,8,12,11)(15,19)(18,20)(21,22,27,25)(23,28,24,30)(26,29,34,33)(31,35,32,36);(6,10)(7,12)(14,17)(18,20)(21,23,27,24)(22,28,25,30)(26,32,34,31)(29,36,33,35);(5,7)(6,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,24,27,23)(22,28,25,30)(26,32,34,31)(29,35,33,36);(1,2)(3,4)(5,8,9,11)(6,7,10,12)(21,25,27,22)(23,30,24,28)(26,33,34,29)(31,36,32,35);(5,7)(6,8)(9,12)(10,11)(13,15,16,19)(14,18,17,20)(21,26)(22,29)(23,31)(24,32)(25,33)(27,34)(28,35)(30,36)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=fg,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5529817);