Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233220 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,14)(7,11,16,20)(8,13,9,12)(15,18,17,19)(21,22,27,32)(23,29,33,36)(24,35,34,28)(25,31,26,30);(1,2)(3,4)(5,7)(6,11)(8,17)(9,15)(10,16)(12,18)(13,19)(14,20)(21,23)(22,28)(24,25)(26,34)(27,33)(29,31)(30,36)(32,35);(2,4)(6,12)(8,9)(11,18)(13,14)(15,17)(19,20)(21,24,25,23,27,34,26,33)(22,29,30,35,32,36,31,28);(5,8,10,9)(6,13,14,12)(7,15,16,17)(11,18,20,19)(23,33)(24,34)(28,35)(29,36)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gh,d^c=di,f^c=fhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 147641);