Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,10)(7,11,12,9)(15,19)(18,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,35)(34,36);(1,2,3,4)(7,12)(9,11)(22,23)(24,31)(25,33)(35,36);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14,17,16)(15,18,19,20)(21,23,26,22)(24,27,31,29)(25,30,33,28)(32,35,34,36);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,20)(17,19)(21,24,26,31)(22,27,23,29)(25,34,33,32)(28,36,30,35);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,20,19,18)(21,25)(22,28)(23,30)(24,32)(26,33)(27,35)(29,36)(31,34)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=gi,d^2=f,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3144418);