Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233265 |
| 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)(7,15)(8,11)(9,13)(10,14)(12,18)(16,20)(17,19)(21,22,26,31)(23,27,33,36)(24,30,25,29)(28,34,35,32);(2,4)(6,11)(7,16)(8,10)(13,14)(17,18)(19,20)(21,23)(22,27)(24,34)(25,32)(26,33)(28,29)(30,35)(31,36);(5,7,9,18)(6,12,13,15)(8,16,10,17)(11,19,14,20)(21,23)(22,28)(24,32)(25,34)(26,33)(27,29)(30,36)(31,35);(6,13)(11,14)(12,15)(19,20)(21,24,26,25)(22,29,31,30)(23,32,33,34)(27,35,36,28)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dhi,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=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140491);