Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,13)(9,18)(10,14)(12,15)(16,20)(17,19)(22,27)(23,31)(24,25)(29,30)(32,35)(33,34);(2,4)(5,7)(6,11)(8,17)(9,15)(10,16)(12,18)(13,19)(14,20)(21,22,26,30)(23,32,34,35)(24,27,25,29)(28,31,36,33);(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,20)(14,19)(21,23,26,34)(22,28,30,36)(24,33,25,31)(27,35,29,32);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,12,20,19)(21,24,26,25)(22,29,30,27)(23,33,34,31)(28,32,36,35)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,g^b=gh,e^c=eh,f^c=fi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140410);