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 | |
4636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10,14,17)(11,18,21,15)(12,16,23,13)(19,24,22,20)(25,26,31,37)(27,35,29,32)(28,34,39,33)(30,38,40,36);(1,3,5,8)(2,6,7,4)(9,11,14,21)(10,15,17,18)(12,22,23,19)(13,20,16,24)(25,27,31,29)(26,32,37,35)(28,38,39,36)(30,33,40,34);(1,3)(2,6)(4,7)(5,8)(12,23)(13,16)(19,22)(20,24)(25,28,31,39)(26,33,37,34)(27,38,29,36)(30,32,40,35);(2,7)(3,8)(9,12)(10,16)(11,19)(13,17)(14,23)(15,20)(18,24)(21,22)(25,28)(26,34)(27,36)(29,38)(30,35)(31,39)(32,40)(33,37);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,22)(13,20)(14,21)(16,24)(17,18)(19,23)(25,29,31,27)(26,35,37,32)(28,36,39,38)(30,34,40,33);(1,4)(2,3)(5,6)(7,8)(9,13,14,16)(10,12,17,23)(11,20,21,24)(15,22,18,19)(25,30,31,40)(26,36,37,38)(27,33,29,34)(28,35,39,32)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841697);