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×#3536 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7018 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)(11,18)(14,20)(17,19)(22,27)(24,32)(25,35)(28,31);(1,2)(3,4)(5,6,9,12)(7,10,15,13)(8,11,16,18)(14,17,20,19)(22,27)(28,31)(29,34)(30,36);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,31)(24,34)(25,30)(26,27)(28,33)(29,32)(35,36);(21,23,26,33)(22,28,27,31)(24,25,32,35)(29,30,34,36);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,24,26,32)(22,29,27,34)(23,25,33,35)(28,30,31,36);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,25)(22,30)(23,32)(24,33)(26,35)(27,36)(28,34)(29,31)Polycyclic presentation:
2^9|b^2=hi,d^2=h,e^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, 7616683);