Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| 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×#3565 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6384 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)(5,6,10,11)(7,14,18,12)(8,9,15,13)(16,19,20,17)(21,22)(23,24)(25,29)(26,30)(27,31)(28,34)(32,33)(35,36);(6,11)(8,15)(12,14)(16,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,34)(32,35)(33,36);(5,7)(6,12)(8,16)(9,17)(10,18)(11,14)(13,19)(15,20)(21,24)(22,28)(23,31)(25,33)(26,29)(27,34)(30,35)(32,36);(5,8,10,15)(6,13,11,9)(7,16,18,20)(12,19,14,17)(21,25)(22,29)(23,32)(24,33)(26,28)(27,35)(30,34)(31,36);(1,3,4,2)(5,7)(6,14)(8,16)(9,19)(10,18)(11,12)(13,17)(15,20)(21,24)(22,23)(25,33)(26,36)(27,34)(28,31)(29,32)(30,35);(5,9,10,13)(6,15,11,8)(7,17,18,19)(12,20,14,16)(21,26,27,32)(22,30,31,33)(23,25,28,35)(24,29,34,36)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534662);