Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994171 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #59102 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,22,27,24)(23,30,34,31)(25,29,26,36)(28,32,35,33);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,33,27,34,25,32)(22,28,29,30,24,35,36,31);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24,27,22)(23,31,34,30)(25,36,26,29)(28,33,35,32);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,29,24,36)(23,32,34,33)(28,30,35,31)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,f^d=fi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7032674);