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 | C22×#540 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7134 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)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(21,22)(23,26)(24,27)(25,30)(28,33)(29,32)(31,35)(34,36);(8,15)(11,13)(16,20)(18,19)(21,22,26,23)(24,29,32,27)(25,28,33,30)(31,35,36,34);(6,7)(11,13)(12,14)(18,19)(22,23)(27,29)(28,30)(34,35);(5,7,10,6)(8,13,15,11)(9,14,17,12)(16,19,20,18)(21,23,26,22)(24,29,32,27)(25,30,33,28)(31,35,36,34);(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=h,e^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, 7533555);