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×#3460 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(21,22,27,29)(23,33,34,28)(24,25,31,30)(26,36,35,32);(7,13)(11,17)(14,19)(18,20)(21,23)(22,28)(24,35)(25,36)(26,31)(27,34)(29,33)(30,32);(6,10)(11,17)(12,15)(18,20)(22,29)(23,34)(24,31)(32,36);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,24,27,31)(22,30,29,25)(23,35,34,26)(28,32,33,36);(5,7,9,13)(6,11,10,17)(8,14,16,19)(12,18,15,20)(21,25,27,30)(22,31,29,24)(23,32,34,36)(26,28,35,33);(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,26,27,35)(22,32,29,36)(23,31,34,24)(25,28,30,33)Polycyclic presentation:
2^9|a^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, 7533568);