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 | |
6984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6,10,12)(7,13,14,8)(9,11,18,20)(15,19,17,16)(21,22,27,29)(23,24,25,28)(26,30,34,36)(31,32,33,35);(1,2)(3,4)(5,7)(6,8)(9,17)(10,14)(11,19)(12,13)(15,18)(16,20)(21,23,27,25)(22,28,29,24)(26,33,34,31)(30,32,36,35);(7,14)(8,13)(15,17)(16,19)(22,29)(23,25)(30,36)(31,33);(5,7)(6,8)(9,15)(10,14)(11,16)(12,13)(17,18)(19,20)(21,24)(22,23)(25,29)(26,32)(27,28)(30,31)(33,36)(34,35);(5,8)(6,7)(9,16)(10,13)(11,15)(12,14)(17,20)(18,19)(21,25,27,23)(22,24,29,28)(26,33,34,31)(30,32,36,35);(5,9,10,18)(6,11,12,20)(7,15,14,17)(8,16,13,19)(21,26,27,34)(22,30,29,36)(23,31,25,33)(24,32,28,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7651027);