Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6,10,13)(7,9,15,11)(8,12,16,19)(14,17,20,18)(21,22,26,28)(23,24,30,32)(25,27,35,34)(29,31,33,36);(1,2,3,4)(8,16)(12,19)(14,20)(17,18)(21,23)(22,24)(25,33)(26,30)(27,36)(28,32)(29,35)(31,34);(5,7,10,15)(6,11,13,9)(8,14,16,20)(12,18,19,17)(21,24,26,32)(22,23,28,30)(25,31,35,36)(27,29,34,33);(5,7)(6,9)(8,14)(10,15)(11,13)(12,17)(16,20)(18,19)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,34)(35,36);(5,8)(6,12)(7,14)(9,17)(10,16)(11,18)(13,19)(15,20)(21,25)(22,27)(23,29)(24,31)(26,35)(28,34)(30,33)(32,36);(5,9)(6,7)(8,17)(10,11)(12,14)(13,15)(16,18)(19,20)(21,22)(23,24)(25,34)(26,28)(27,35)(29,36)(30,32)(31,33)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7944185);