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 | |
6878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6)(7,15)(8,14)(9,12)(10,16)(11,13)(17,18)(19,20)(21,22,25,28)(23,29,31,26)(24,27,33,32)(30,35,36,34);(1,2)(3,4)(5,7,11,9)(6,12,13,15)(8,17,16,19)(10,18,14,20)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(6,13)(8,16)(12,15)(17,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(5,8,11,16)(6,14,13,10)(7,17,9,19)(12,20,15,18);(5,9,11,7)(6,15,13,12)(8,19,16,17)(10,20,14,18)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,10,11,14)(6,16,13,8)(7,18,9,20)(12,19,15,17)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7645009);