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 | |
4368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(26,32)(27,28)(33,40)(34,36);(1,3)(2,6)(4,7)(5,8)(11,17)(15,21)(18,23)(22,24)(25,26,31,32)(27,29,28,35)(30,33,39,40)(34,37,36,38);(1,3,5,8)(2,6,7,4)(10,14)(15,21)(16,19)(22,24)(25,27,31,28)(26,29,32,35)(30,34,39,36)(33,37,40,38);(3,8)(4,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,31,27)(26,29,32,35)(30,36,39,34)(33,37,40,38);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29)(26,28)(27,32)(30,38)(31,35)(33,34)(36,40)(37,39);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,39)(26,33,32,40)(27,34,28,36)(29,37,35,38)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10224042);