Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,9)(7,11)(15,18)(16,20)(21,22,26,27)(23,30,33,28)(24,29,34,25)(31,35,36,32);(5,6)(7,12)(8,9)(10,11)(13,14,17,19)(15,16,18,20)(22,27)(24,34)(28,30)(31,36);(1,2,3,4)(6,9)(10,12)(13,15,17,18)(14,16,19,20)(21,23,26,33)(22,28,27,30)(24,31,34,36)(25,35,29,32);(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,24)(22,29)(23,31)(25,27)(26,34)(28,32)(30,35)(33,36);(5,6)(7,10)(8,9)(11,12)(21,25,26,29)(22,24,27,34)(23,32,33,35)(28,36,30,31)Polycyclic presentation:
2^9|a^2=fh,b^2=hi,c^2=g,d^2=h,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4293591);