Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,8,15,11)(9,12)(10,13)(14,16,20,18)(17,19)(21,22,26,30,27,23,25,29)(24,28,34,36,32,31,33,35);(2,4)(5,7)(6,11)(8,13)(9,14)(10,15)(12,18)(16,19)(17,20)(21,23,27,22)(24,31,32,28)(25,29,26,30)(33,35,34,36);(5,8,10,11)(6,7,13,15)(9,16,17,18)(12,14,19,20)(21,24)(22,28)(23,31)(25,33)(26,34)(27,32)(29,35)(30,36);(5,9,10,17)(6,12,13,19)(7,14,15,20)(8,16,11,18)(24,32)(28,31)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh,g^c=gh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93224);