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 | |
4700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,20)(12,18)(13,17)(14,16)(15,19)(21,23)(22,24)(25,26)(27,29)(28,37)(30,34)(31,35)(32,36)(33,40)(38,39);(1,3,4,8)(2,5,6,7)(9,11,15,14)(10,16,19,20)(12,21,13,22)(17,23,18,24)(25,27,32,31)(26,29,36,35)(28,38,40,34)(30,37,39,33);(1,3)(2,5)(4,8)(6,7)(9,12,15,13)(10,17,19,18)(11,21,14,22)(16,23,20,24)(25,28,32,40)(26,33,36,37)(27,34,31,38)(29,39,35,30);(1,3)(2,5)(4,8)(6,7)(9,13,15,12)(10,18,19,17)(11,21,14,22)(16,23,20,24)(25,29)(26,27)(28,39)(30,40)(31,36)(32,35)(33,34)(37,38);(1,2)(3,5)(4,6)(7,8)(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,24)(22,23)(25,30,32,39)(26,34,36,38)(27,33,31,37)(28,29,40,35);(9,14,15,11)(10,16,19,20)(12,22,13,21)(17,23,18,24)(25,31,32,27)(26,35,36,29)(28,38,40,34)(30,37,39,33)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gi,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8412500);