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 | C24 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6272 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)(7,14)(8,11)(9,17)(10,13)(12,18)(15,19)(16,20)(22,26)(24,32)(27,29)(30,36);(1,3,4,2)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,7,10,8)(6,11,13,14)(9,15,18,16)(12,19,17,20)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,6)(7,11)(8,14)(9,12)(10,13)(15,19)(16,20)(17,18)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35);(5,8,10,7)(6,11,13,14)(9,16,18,15)(12,19,17,20);(5,9)(6,12)(7,15)(8,16)(10,18)(11,19)(13,17)(14,20)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,d^2=g,e^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, 7533896);