Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
4373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,20)(12,18)(13,14)(15,17)(16,21)(19,23)(22,24)(25,26,31,28)(27,32,34,30)(29,33,39,36)(35,40,37,38);(1,2,5,6)(3,7,4,8)(9,11,15,21)(10,16,17,20)(12,13,22,23)(14,18,19,24)(25,27,31,34)(26,32,28,30)(29,37,39,35)(33,38,36,40);(1,3,5,4)(2,7,6,8)(10,17)(11,21)(12,22)(14,19)(27,34)(30,32)(35,37)(38,40);(9,12,15,22)(10,18,17,24)(11,13,21,23)(14,16,19,20)(25,28,31,26)(27,30,34,32)(29,33,39,36)(35,40,37,38);(1,4,5,3)(2,8,6,7)(9,13)(10,19)(11,12)(14,17)(15,23)(16,18)(20,24)(21,22)(25,29,31,39)(26,33,28,36)(27,35,34,37)(30,38,32,40);(9,14,15,19)(10,13,17,23)(11,18,21,24)(12,16,22,20)(25,30)(26,27)(28,34)(29,38)(31,32)(33,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7705259);