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:
(1,2,5,8)(3,6,4,7)(9,10,14,18)(11,12,16,15)(13,17,23,24)(19,20,21,22)(25,26,30,33)(27,28,35,31)(29,32,39,40)(34,36,37,38);(1,2)(3,6)(4,7)(5,8)(9,11,14,16)(10,15,18,12)(13,21,23,19)(17,20,24,22)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,39)(38,40);(11,16)(12,15)(19,21)(20,22)(25,27,30,35)(26,31,33,28)(29,34,39,37)(32,38,40,36);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,40)(37,39);(1,3,5,4)(2,6,8,7)(9,12,14,15)(10,16,18,11)(13,22,23,20)(17,19,24,21)(25,28,30,31)(26,27,33,35)(29,38,39,36)(32,37,40,34);(1,4,5,3)(2,7,8,6)(9,13,14,23)(10,17,18,24)(11,19,16,21)(12,20,15,22)(25,29,30,39)(26,32,33,40)(27,34,35,37)(28,36,31,38)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7876875);