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 | |
6694 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)(5,6,11,13)(7,17,18,12)(8,10,16,14)(9,19,20,15)(24,32)(28,35)(31,34)(33,36);(1,2)(3,4)(5,7)(6,12)(8,15)(9,14)(10,20)(11,18)(13,17)(16,19)(21,22,27,30)(23,29,26,25)(24,28,32,35)(31,36,34,33);(6,13)(7,18)(8,16)(9,20)(21,23,27,26)(22,25,30,29)(24,31,32,34)(28,33,35,36);(5,8,11,16)(6,14,13,10)(7,15,18,19)(9,17,20,12)(21,24)(22,28)(23,31)(25,33)(26,34)(27,32)(29,36)(30,35);(5,9,11,20)(6,15,13,19)(7,14,18,10)(8,17,16,12)(21,25)(22,23)(24,33)(26,30)(27,29)(28,31)(32,36)(34,35);(5,10,11,14)(6,16,13,8)(7,9,18,20)(12,15,17,19)(21,26,27,23)(22,29,30,25)(24,34,32,31)(28,36,35,33)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7893552);