Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| 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 | |
2070 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)(5,6,10,12)(7,9,8,11)(13,14,17,18)(15,19,20,16)(21,22)(23,29)(24,26)(25,28)(27,30)(31,35)(32,33)(34,36);(5,7,10,8)(6,11,12,9)(13,15,17,20)(14,16,18,19)(22,24)(23,30)(25,33)(34,35);(5,7,10,8)(6,9,12,11)(13,16)(14,15)(17,19)(18,20)(21,23)(22,27)(24,29)(25,31)(26,30)(28,34)(32,35)(33,36);(1,2)(3,4)(5,8,10,7)(6,11,12,9)(13,14)(15,16)(17,18)(19,20)(21,24,26,22)(23,29,30,27)(25,32,33,28)(31,35,36,34);(5,9,10,11)(6,7,12,8)(13,15)(14,16)(17,20)(18,19)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fh,d^2=fgh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1947048);