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 | |
6774 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)(7,17)(8,13)(9,14)(10,15)(11,16)(12,18)(19,20)(21,22,27,32)(23,28,35,29)(24,34,33,30)(25,31,36,26);(1,2,3,4)(5,7)(6,12)(8,10)(9,15)(11,18)(13,20)(14,19)(16,17)(21,23,27,35)(22,28,32,29)(24,25,33,36)(26,34,31,30);(7,18)(10,19)(12,17)(15,20)(21,23,27,35)(22,29,32,28)(24,25,33,36)(26,30,31,34);(5,8)(6,13)(7,10)(9,16)(11,14)(12,20)(15,17)(18,19)(21,24)(22,30)(23,25)(26,28)(27,33)(29,31)(32,34)(35,36);(5,9,11,13)(6,14,16,8)(7,15,18,20)(10,12,19,17)(21,25)(22,31)(23,33)(24,35)(26,32)(27,36)(28,30)(29,34);(5,10)(6,15)(7,8)(9,12)(11,19)(13,17)(14,18)(16,20)(21,26,27,31)(22,25,32,36)(23,34,35,30)(24,28,33,29)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7558981);