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 | |
4330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,19,21,15)(12,17,22,20)(13,23,24,18)(25,26)(27,33)(28,30)(29,32)(31,35)(34,39)(36,37)(38,40);(1,2)(3,7)(4,5)(6,8)(9,11,14,21)(10,15,16,19)(12,18,22,23)(13,17,24,20)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(2,5)(6,7)(10,16)(12,22)(15,19)(18,23)(29,37)(32,36)(34,40)(38,39);(2,5)(6,7)(9,11)(10,15)(12,18)(13,20)(14,21)(16,19)(17,24)(22,23)(25,28,30,26)(27,31,35,33)(29,32,37,36)(34,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,18)(13,19)(14,22)(15,24)(16,20)(21,23)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39);(1,3)(2,6)(4,8)(5,7)(9,13,14,24)(10,18,16,23)(11,20,21,17)(12,19,22,15)(25,27)(26,33)(28,31)(29,34)(30,35)(32,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231255);