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×#3554 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6626 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)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(22,27)(23,30)(25,33)(32,35);(7,13)(11,17)(14,19)(18,20)(21,22,26,27)(23,24,30,28)(25,29,33,34)(31,32,36,35);(6,10)(11,17)(12,15)(18,20)(21,23,26,30)(22,28,27,24)(25,31,33,36)(29,35,34,32);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,23)(22,24)(25,31)(26,30)(27,28)(29,32)(33,36)(34,35);(1,3,4,2)(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,20)(16,19)(21,24)(22,23)(25,32)(26,28)(27,30)(29,31)(33,35)(34,36);(5,8,9,16)(6,12,10,15)(7,14,13,19)(11,18,17,20)(21,25,26,33)(22,29,27,34)(23,31,30,36)(24,32,28,35)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,e^2=i,f^2=h,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, 7533723);