Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233265 |
| 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×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
948 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)(7,11)(8,13)(9,18)(10,14)(12,15)(16,20)(17,19)(21,22,26,30)(23,31,33,35)(24,27,25,29)(28,32,36,34);(1,2)(3,4)(5,7)(6,11)(8,17)(9,15)(10,16)(12,18)(13,19)(14,20)(22,27)(23,32)(24,25)(28,36)(29,30)(33,34);(2,4)(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,20)(14,19)(21,23,24,34,26,33,25,32)(22,28,29,31,30,36,27,35);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,12,20,19)(23,33)(28,36)(31,35)(32,34)Polycyclic presentation:
2^9|a^2=h,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=dhi,e^b=eh,f^b=fh,g^b=gh,d^c=dh,f^c=fhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 147696);