Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,9)(8,12)(10,11)(13,14,18,20)(15,16,19,17)(22,27)(23,30)(29,32)(31,36);(6,11)(7,8)(13,15)(14,17)(16,20)(18,19)(21,22,26,27)(23,28,30,25)(24,32,34,29)(31,35,36,33);(5,7,10,8)(6,12,11,9)(13,16,18,17)(14,19,20,15)(21,23)(22,28)(24,31)(25,27)(26,30)(29,33)(32,35)(34,36);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(13,15)(14,16)(17,20)(18,19)(21,24,26,34)(22,29,27,32)(23,31,30,36)(25,35,28,33);(5,9,10,12)(6,8,11,7)(13,17,18,16)(14,19,20,15)(21,25,26,28)(22,23,27,30)(24,33,34,35)(29,36,32,31)Polycyclic presentation:
2^9|a^2=fgi,b^2=fh,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4775305);