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 | |
1320 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)(5,6,11,15)(7,16,18,19)(8,12,20,17)(9,14,10,13)(21,22,26,29)(23,31,34,35)(24,27,25,30)(28,32,36,33);(2,4)(5,7)(6,8)(9,12)(10,17)(11,18)(13,19)(14,16)(15,20)(22,27)(24,25)(28,35)(29,30)(31,36)(32,33);(5,8,11,20)(6,12,15,17)(7,13,18,14)(9,19,10,16)(21,23,26,34)(22,28,29,36)(24,33,25,32)(27,35,30,31);(5,9,11,10)(6,13,15,14)(7,12,18,17)(8,19,20,16)(22,29)(27,30)(28,36)(31,35)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=hi,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, 140624);