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 | |
7151 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)(8,16)(11,18)(14,20)(17,19)(21,22,27,31)(23,32,34,28)(24,26,30,35)(25,29,33,36);(1,2)(3,4)(5,6)(7,12)(8,11)(9,13)(10,15)(14,19)(16,18)(17,20);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,28)(24,33)(25,30)(26,29)(27,34)(31,32)(35,36);(5,8)(6,11)(7,14)(9,16)(10,17)(12,19)(13,18)(15,20)(21,24,27,30)(22,26,31,35)(23,33,34,25)(28,29,32,36);(5,7)(6,12)(8,14)(9,15)(10,13)(11,19)(16,20)(17,18)(21,25,27,33)(22,29,31,36)(23,30,34,24)(26,32,35,28);(5,6)(7,10)(8,11)(9,13)(12,15)(14,17)(16,18)(19,20)(21,26)(22,30)(23,29)(24,31)(25,28)(27,35)(32,33)(34,36)Polycyclic presentation:
2^9|a^2=gh,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, 7563559);