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 | |
7076 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)(5,6,11,15)(7,16,18,12)(8,10,14,19)(9,13,17,20)(24,32)(27,34)(31,35)(33,36);(1,2)(3,4)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,34)(33,35);(5,7)(6,12)(8,17)(9,14)(10,13)(11,18)(15,16)(19,20)(21,23,26,28)(22,25,29,30)(24,31,32,35)(27,33,34,36);(5,8,11,14)(6,10,15,19)(7,17,18,9)(12,13,16,20)(21,24)(22,27)(23,31)(25,33)(26,32)(28,35)(29,34)(30,36);(5,9,11,17)(6,13,15,20)(7,14,18,8)(10,16,19,12)(21,22)(23,25)(24,27)(26,29)(28,30)(31,33)(32,34)(35,36);(5,10)(6,14)(7,13)(8,15)(9,12)(11,19)(16,17)(18,20)(21,25)(22,28)(23,29)(24,33)(26,30)(27,35)(31,34)(32,36)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7564187);