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 | |
4482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,18,12,15)(13,21,22,16)(19,23,20,24)(25,26,30,35)(27,31,36,32)(28,33,38,39)(29,37,40,34);(1,2)(3,7)(4,5)(6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,27)(26,31)(28,37)(29,39)(30,36)(32,35)(33,40)(34,38);(2,5)(6,7)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(25,27,30,36)(26,32,35,31)(28,34,38,37)(29,33,40,39);(2,5)(6,7)(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,24)(21,22)(25,28)(26,33)(27,34)(29,31)(30,38)(32,40)(35,39)(36,37);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,35)(37,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,22)(10,16,17,21)(11,19,12,20)(15,23,18,24)(25,29)(26,34)(27,33)(28,31)(30,40)(32,38)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=g,f^2=g,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, 10228000);