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 | |
4311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,24)(22,23)(25,26,32,29)(27,35,31,33)(28,37,30,34)(36,39,38,40);(1,3)(2,6)(4,8)(5,7)(9,11,15,14)(10,16,19,20)(12,21,13,22)(17,23,18,24)(25,27,32,31)(26,33,29,35)(28,36,30,38)(34,39,37,40);(1,2)(3,4)(5,8)(6,7)(9,12,15,13)(10,17,19,18)(11,21,14,22)(16,23,20,24)(25,28,32,30)(26,34,29,37)(27,36,31,38)(33,39,35,40);(1,4)(2,7)(3,8)(5,6)(9,13,15,12)(10,18,19,17)(11,22,14,21)(16,24,20,23)(25,29,32,26)(27,35,31,33)(28,34,30,37)(36,39,38,40);(1,4)(2,3)(5,6)(7,8)(9,14,15,11)(10,16,19,20)(12,22,13,21)(17,23,18,24)(25,30,32,28)(26,34,29,37)(27,36,31,38)(33,40,35,39);(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,18)(14,20)(15,19)(21,23)(22,24)(25,31,32,27)(26,35,29,33)(28,38,30,36)(34,40,37,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9623334);