Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1901 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)(5,6,9,11)(7,12,8,10)(13,14,17,16)(15,19,20,18)(21,22,25,28)(23,29,30,26)(24,27,32,34)(31,35,36,33);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14,17,16)(15,18,20,19)(23,30)(24,32)(26,29)(27,34);(1,2,3,4)(6,11)(10,12)(13,14,17,16)(15,18,20,19)(21,23,25,30)(22,26,28,29)(24,31,32,36)(27,33,34,35);(5,8,9,7)(6,12,11,10)(13,15,17,20)(14,18,16,19)(21,24)(22,27)(23,31)(25,32)(26,33)(28,34)(29,35)(30,36);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,16,17,14)(15,19,20,18)(21,22)(23,26)(24,27)(25,28)(29,30)(31,33)(32,34)(35,36)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=gi,d^2=i,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2508424);