Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6)(7,10)(8,12)(9,11)(21,22,25,26)(23,27,29,34)(24,31,32,28)(30,36,33,35);(2,3)(5,7,9,12)(6,10,11,8)(13,14,16,18)(15,17,19,20)(22,26)(23,29)(24,32)(35,36);(1,2,4,3)(5,8)(6,7)(9,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,27)(24,31)(25,26)(28,32)(29,34)(30,36)(33,35);(5,8)(6,7)(9,10)(11,12)(21,23,25,29)(22,27,26,34)(24,33,32,30)(28,36,31,35);(1,3,4,2)(5,7)(6,8)(9,12)(10,11)(13,15,16,19)(14,17,18,20)(21,24,25,32)(22,28,26,31)(23,30,29,33)(27,35,34,36)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fgi,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3981834);