Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246847 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3219 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,14)(8,16)(9,10)(17,18)(19,20)(22,27)(24,32)(25,33)(28,36)(30,34)(31,35);(1,2)(3,4)(5,6,11,7)(8,13,17,15)(9,14,10,12)(16,19,18,20)(21,22,26,30)(23,28,29,24)(25,34,33,27)(31,32,35,36);(2,4)(5,7,11,6)(8,15,17,13)(9,14,10,12)(16,19,18,20)(21,23,26,29)(22,28,30,24)(25,35,33,31)(27,32,34,36);(5,8)(6,13)(7,15)(9,18)(10,16)(11,17)(12,19)(14,20)(21,24,26,28)(22,29,30,23)(25,36,33,32)(27,31,34,35)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 154088);