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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,8,10)(7,11,12,9)(15,19)(18,20)(21,22,25,26)(23,29,30,27)(24,28,33,32)(31,35,36,34);(5,6,8,10)(7,9,12,11)(14,17)(18,20)(22,26)(23,30)(28,32)(31,36);(5,6)(7,9)(8,10)(11,12)(13,14,16,17)(15,18,19,20)(21,22)(23,27)(24,32)(25,26)(28,33)(29,30)(31,35)(34,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(21,23,25,30)(22,27,26,29)(24,31,33,36)(28,34,32,35);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=gh,d^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, 4514763);