Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2362 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)(5,6,9,7)(8,11,12,10)(13,14)(15,19)(16,18)(17,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,36)(32,34)(33,35);(8,12)(10,11)(13,15)(14,19)(16,17)(18,20)(21,22,25,28)(23,26,31,29)(24,27,34,32)(30,35,33,36);(6,7)(10,11)(14,16)(15,20)(21,22)(23,26)(24,32)(25,28)(27,34)(29,31)(30,36)(33,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,17,20,19)(21,23,25,31)(22,26,28,29)(24,33,34,30)(27,36,32,35);(5,8)(6,10)(7,11)(9,12)(13,17)(14,15)(16,20)(18,19)(21,24)(22,27)(23,30)(25,34)(26,35)(28,32)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4039268);