Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| 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 | |
1336 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)(5,6,10,13)(7,12,17,19)(8,11,9,14)(15,18,16,20)(21,22,27,31)(23,32,24,35)(25,30,26,29)(28,33,36,34);(2,4)(6,11)(7,15)(8,9)(13,14)(16,17)(18,20)(21,23)(22,28)(24,27)(25,33)(26,34)(29,35)(30,32)(31,36);(5,7,10,17)(6,12,13,19)(8,15,9,16)(11,18,14,20)(21,24,27,23)(22,28,31,36)(25,34,26,33)(29,35,30,32);(6,13)(11,14)(12,19)(18,20)(21,25,27,26)(22,29,31,30)(23,33,24,34)(28,35,36,32)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140616);