Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3516 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,4,3)(5,6)(7,10)(8,11)(9,14)(12,17)(13,16)(15,19)(18,20)(21,22,26,23)(24,29,32,27)(25,28,33,30)(31,35,36,34);(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18)(22,23)(27,29)(28,30)(34,35);(1,3,4,2)(6,7)(11,13)(12,14)(18,19)(21,23,26,22)(24,29,32,27)(25,30,33,28)(31,35,36,34);(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,25,26,33)(22,28,23,30)(24,31,32,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533617);