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 | |
4496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,13)(12,15)(14,20)(16,18)(17,22)(19,23)(21,24)(25,26,32,30)(27,35,31,33)(28,34,29,38)(36,40,37,39);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24)(25,27,32,31)(26,33,30,35)(28,36,29,37)(34,39,38,40);(1,3,5,6)(2,4,7,8)(10,12)(11,18)(17,20)(19,24)(25,28,32,29)(26,34,30,38)(27,36,31,37)(33,39,35,40);(1,4)(2,6)(3,7)(5,8)(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23)(25,29,32,28)(26,34,30,38)(27,37,31,36)(33,39,35,40);(1,2)(3,4)(5,7)(6,8)(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23)(25,30,32,26)(27,35,31,33)(28,38,29,34)(36,40,37,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,22)(10,17,12,20)(11,19,18,24)(13,21,16,23)(25,31,32,27)(26,33,30,35)(28,37,29,36)(34,39,38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10010655);