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 | |
4787 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)(9,10,14,15)(11,16,12,19)(13,17,21,22)(18,23,20,24)(26,27)(28,35)(29,37)(38,39);(1,2,4,6)(3,5,8,7)(10,15)(16,19)(17,22)(23,24)(25,26,30,27)(28,33,35,31)(29,32,37,34)(36,39,40,38);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,16,15,19)(13,18,21,20)(17,23,22,24)(25,26,30,27)(28,31,35,33)(29,32,37,34)(36,38,40,39);(3,8)(5,7)(9,12,14,11)(10,16,15,19)(13,18,21,20)(17,24,22,23)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,19)(13,17)(14,15)(18,23)(20,24)(21,22)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,13,14,21)(10,17,15,22)(11,18,12,20)(16,23,19,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=g,f^2=g,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, 10315296);