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 |
| prime radical | C2×#3541 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6782 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)(7,14)(9,17)(11,12)(13,16)(24,32)(27,34)(30,36)(33,35);(1,2)(3,4)(5,6,10,8)(7,12,14,11)(9,13,17,16)(15,19,20,18)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(5,7,10,14)(6,11,8,12)(9,15,17,20)(13,18,16,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,8,10,6)(7,11,14,12)(9,16,17,13)(15,18,20,19);(5,7)(6,12)(8,11)(9,15)(10,14)(13,19)(16,18)(17,20)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,9)(6,13)(7,15)(8,16)(10,17)(11,18)(12,19)(14,20)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^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, 7616501);