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×#3535 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,4,3)(5,6)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(21,22,27,29)(23,32,33,28)(24,34,31,26)(25,30,35,36);(7,13)(11,17)(14,19)(18,20)(21,23)(22,28)(24,25)(26,30)(27,33)(29,32)(31,35)(34,36);(6,10)(11,17)(12,15)(18,20)(22,29)(23,33)(24,31)(30,36);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,24,27,31)(22,26,29,34)(23,25,33,35)(28,30,32,36);(1,3,4,2)(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,20)(16,19)(21,25)(22,30)(23,31)(24,33)(26,32)(27,35)(28,34)(29,36);(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,26)(22,31)(23,30)(24,29)(25,32)(27,34)(28,35)(33,36)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=gi,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, 7533656);