Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,8)(9,11)(10,12)(15,19)(18,20)(21,22)(23,29)(24,32)(25,26)(27,31)(28,34)(30,35)(33,36);(5,7,10,11)(6,9,12,8)(14,17)(18,20)(22,26)(27,29)(28,32)(35,36);(5,8,10,9)(6,11,12,7)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,35)(32,34)(33,36);(1,2)(3,4)(5,9,10,8)(6,11,12,7)(21,23,25,31)(22,27,26,29)(24,33,34,30)(28,36,32,35);(5,6)(7,9)(8,11)(10,12)(13,15)(14,18)(16,19)(17,20)(21,24,25,34)(22,28,26,32)(23,30,31,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4774312);