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 | |
6828 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)(7,12)(8,16)(9,14)(10,11)(13,18)(15,17)(19,20)(25,33)(29,34)(30,35)(32,36);(1,2)(3,4)(21,22,26,27)(23,24,28,31)(25,29,33,34)(30,32,35,36);(6,11)(7,14)(13,16)(15,20)(22,27)(24,31)(29,34)(32,36);(5,7,10,14)(6,12,11,9)(8,15,18,20)(13,19,16,17)(21,23,26,28)(22,24,27,31)(25,30,33,35)(29,32,34,36);(5,8,10,18)(6,13,11,16)(7,15,14,20)(9,17,12,19)(21,24)(22,28)(23,27)(25,32)(26,31)(29,35)(30,34)(33,36);(5,9,10,12)(6,14,11,7)(8,17,18,19)(13,20,16,15)(21,25)(22,29)(23,30)(24,32)(26,33)(27,34)(28,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618005);