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 | C2×#3515 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6550 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)(5,6,10,14)(7,15,17,11)(8,18,16,12)(9,13,19,20)(21,22,27,31)(23,30,26,28)(24,25,29,36)(32,35,34,33);(1,3,4,2)(5,7)(6,11)(8,19)(9,16)(10,17)(12,20)(13,18)(14,15);(5,8,10,16)(6,12,14,18)(7,9,17,19)(11,13,15,20)(21,23,27,26)(22,28,31,30)(24,32,29,34)(25,33,36,35);(5,7)(6,11)(8,9)(10,17)(12,13)(14,15)(16,19)(18,20)(21,24)(22,25)(23,32)(26,34)(27,29)(28,33)(30,35)(31,36);(5,9)(6,13)(7,16)(8,17)(10,19)(11,18)(12,15)(14,20)(21,25)(22,29)(23,33)(24,31)(26,35)(27,36)(28,34)(30,32);(5,6)(7,11)(8,12)(9,13)(10,14)(15,17)(16,18)(19,20)(21,26,27,23)(22,30,31,28)(24,34,29,32)(25,35,36,33)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,f^2=gh,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, 7533965);