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 | |
6744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6)(7,9)(8,16)(10,11)(12,14)(13,17)(15,18)(19,20)(21,22)(23,25)(24,32)(26,30)(27,28)(29,35)(31,36)(33,34);(1,2,3,4)(5,7)(6,9)(8,15)(10,14)(11,12)(13,19)(16,18)(17,20)(22,28)(23,26)(29,32)(31,34);(6,11)(7,14)(8,17)(18,19)(21,23,27,26)(22,25,28,30)(24,31,35,34)(29,33,32,36);(5,7)(6,12)(8,15)(9,11)(10,14)(13,18)(16,19)(17,20)(21,24,27,35)(22,29,28,32)(23,31,26,34)(25,33,30,36);(5,8)(6,13)(7,15)(9,19)(10,17)(11,16)(12,18)(14,20)(21,25)(22,23)(24,33)(26,28)(27,30)(29,31)(32,34)(35,36);(5,9,10,12)(6,7,11,14)(8,18,17,19)(13,20,16,15)(21,26,27,23)(22,30,28,25)(24,34,35,31)(29,36,32,33)Polycyclic presentation:
2^9|b^2=ghi,c^2=hi,d^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7894694);