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 | |
6408 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)(21,22,26,28)(23,27,29,24)(25,30,34,36)(31,35,33,32);(1,2)(3,4)(5,7,11,9)(6,12,13,15)(8,17,16,19)(10,18,14,20)(21,23,26,29)(22,27,28,24)(25,33,34,31)(30,32,36,35);(6,13)(8,16)(12,15)(17,19)(22,28)(23,29)(30,36)(31,33);(5,8,11,16)(6,14,13,10)(7,17,9,19)(12,20,15,18)(21,24,26,27)(22,29,28,23)(25,32,34,35)(30,33,36,31);(5,9,11,7)(6,15,13,12)(8,19,16,17)(10,20,14,18)(21,22)(23,27)(24,29)(25,30)(26,28)(31,35)(32,33)(34,36);(5,10,11,14)(6,16,13,8)(7,18,9,20)(12,19,15,17)(21,25,26,34)(22,30,28,36)(23,31,29,33)(24,32,27,35)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650389);