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×#540 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6302 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)(8,15)(9,11)(12,16)(13,19)(14,17)(18,20)(21,22,26,29)(23,27,31,25)(24,32,34,28)(30,33,36,35);(5,7,9,14)(6,10,11,17)(8,13,16,20)(12,18,15,19)(21,23,26,31)(22,25,29,27)(24,30,34,36)(28,33,32,35);(6,11)(10,17)(12,15)(18,19)(21,23)(22,27)(24,30)(25,29)(26,31)(28,35)(32,33)(34,36);(5,6)(7,10)(8,12)(9,11)(13,18)(14,17)(15,16)(19,20)(21,24,26,34)(22,28,29,32)(23,30,31,36)(25,33,27,35);(1,3,4,2)(5,7)(6,10)(8,13)(9,14)(11,17)(12,18)(15,19)(16,20);(5,8)(6,12)(7,13)(9,16)(10,18)(11,15)(14,20)(17,19)(21,25,26,27)(22,23,29,31)(24,33,34,35)(28,30,32,36)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=g,e^2=gi,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, 7533685);