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 | |
4552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,17)(14,18)(15,16)(19,20)(21,23)(22,24)(25,26,30,31)(27,28,35,32)(29,37,39,33)(34,40,38,36);(1,3,7,6)(2,4,8,5)(9,10,14,18)(11,15,16,12)(13,17,22,24)(19,23,21,20)(26,31)(28,32)(33,37)(36,40);(1,4,7,5)(2,6,8,3)(25,27)(26,32)(28,31)(29,34)(30,35)(33,40)(36,37)(38,39);(1,4,7,5)(2,3,8,6)(9,11)(10,15)(12,18)(13,21)(14,16)(17,20)(19,22)(23,24)(25,27)(26,32)(28,31)(29,38)(30,35)(33,36)(34,39)(37,40);(1,5,7,4)(2,3,8,6)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,21,24,19)(25,28,30,32)(26,27,31,35)(29,36,39,40)(33,34,37,38);(1,6,7,3)(2,4,8,5)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24)(25,29,30,39)(26,33,31,37)(27,34,35,38)(28,36,32,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=i,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10352137);