Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233220 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 | |
976 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)(7,10)(8,16)(9,13)(11,15)(12,18)(14,19)(17,20)(21,22)(23,31)(24,25)(26,29)(27,30)(28,33)(32,34)(35,36);(1,2)(3,4)(5,7)(6,10)(8,15)(9,14)(11,16)(12,20)(13,19)(17,18)(21,23,27,35)(22,28,30,32)(24,31,29,36)(25,34,26,33);(2,4)(5,7)(6,11)(8,17)(9,14)(10,16)(12,19)(13,20)(15,18)(21,24,25,22,27,29,26,30)(23,32,33,36,35,28,34,31);(7,14)(10,19)(11,20)(15,17)(21,25,27,26)(22,24,30,29)(23,33,35,34)(28,36,32,31)Polycyclic presentation:
2^9|b^2=h,c^2=e,d^2=h,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, 147626);