Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233501 |
| 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 | C24 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1228 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)(5,6,11,16)(7,17,13,20)(8,18,12,19)(9,14,10,15)(21,22)(23,31)(24,30)(25,29)(26,28)(27,33)(32,36)(34,35);(2,4)(5,7,11,13)(6,12,16,8)(9,19,10,18)(14,17,15,20)(23,32)(24,25)(27,35)(29,30)(31,36)(33,34);(5,8,11,12)(6,13,16,7)(9,17,10,20)(14,19,15,18)(21,23,26,34)(22,27,28,36)(24,32,25,33)(29,35,30,31);(5,9,11,10)(6,14,16,15)(7,18,13,19)(8,17,12,20)(22,28)(27,36)(29,30)(31,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140711);