Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
2044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,10)(7,11,12,9)(14,18)(16,20)(21,22,26,24)(23,29,30,27)(25,28,33,32)(31,35,36,34);(1,2,3,4)(7,12)(9,11)(14,18)(15,19)(22,24)(23,30)(25,33)(34,35);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14,17,18)(15,16,19,20)(21,23,26,30)(22,27,24,29)(25,31,33,36)(28,34,32,35);(5,7)(6,9)(8,12)(10,11)(13,15,17,19)(14,16,18,20)(21,24,26,22)(23,27,30,29)(25,28,33,32)(31,35,36,34);(5,6)(7,9)(8,10)(11,12)(13,16)(14,19)(15,18)(17,20)(21,25,26,33)(22,28,24,32)(23,31,30,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,c^2=gi,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3093386);