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 | |
6790 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)(8,16)(12,17)(14,20)(18,19)(22,25)(23,30)(26,29)(32,34);(1,2)(3,4)(6,10)(11,13)(12,17)(18,19)(21,22,27,25)(23,31,30,28)(24,29,35,26)(32,36,34,33);(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18)(21,23,27,30)(22,28,25,31)(24,34,35,32)(26,33,29,36);(21,24,27,35)(22,29,25,26)(23,32,30,34)(28,33,31,36);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,25,27,22)(23,28,30,31)(24,26,35,29)(32,33,34,36);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)(21,26)(22,24)(23,33)(25,35)(27,29)(28,34)(30,36)(31,32)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=h,e^2=i,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, 7618617);