Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| 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×#41 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,16)(9,13)(10,15)(12,17)(14,18)(19,20)(21,22)(23,32)(24,35)(25,29)(26,36)(27,31)(28,34)(30,33);(21,23,27,34)(22,28,31,32)(24,33,29,26)(25,30,35,36);(5,7)(6,10)(8,14)(9,15)(11,13)(12,18)(16,19)(17,20)(21,22,27,31)(23,32,34,28)(24,25,29,35)(26,30,33,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,19)(16,18)(17,20)(21,24)(22,25)(23,33)(26,34)(27,29)(28,30)(31,35)(32,36);(5,6)(7,10)(8,12)(9,13)(11,15)(14,18)(16,17)(19,20)(21,25)(22,29)(23,30)(24,31)(26,28)(27,35)(32,33)(34,36);(5,8,9,17)(6,12,13,16)(7,14,15,20)(10,18,11,19)(21,26,27,33)(22,30,31,36)(23,24,34,29)(25,28,35,32)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533831);