Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,16)(15,18,20,19)(22,26)(24,31)(27,32)(30,35);(5,6,8,10)(7,9,12,11)(14,16)(18,19)(23,29)(27,32)(30,35)(34,36);(7,12)(9,11)(13,14,17,16)(15,19,20,18)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,20)(14,18,16,19)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,19,20,18)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)Polycyclic presentation:
2^9|a^2=fi,b^2=fh,c^2=h,d^2=h,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1764159);