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 | |
4646 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)(3,7)(4,5)(6,8)(9,10,14,18)(11,13,17,15)(12,16,20,24)(19,22,21,23)(25,26,30,31)(27,33,35,29)(28,36,37,32)(34,39,40,38);(3,8)(6,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(25,27,30,35)(26,29,31,33)(28,34,37,40)(32,39,36,38);(1,3,5,8)(2,6,4,7)(11,17)(12,20)(13,15)(16,24)(26,31)(28,37)(29,33)(34,40);(3,8)(6,7)(9,11,14,17)(10,15,18,13)(12,21,20,19)(16,22,24,23)(25,28)(26,32)(27,34)(29,39)(30,37)(31,36)(33,38)(35,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,20)(10,16,18,24)(11,19,17,21)(13,22,15,23)(25,29)(26,27)(28,38)(30,33)(31,35)(32,40)(34,36)(37,39);(1,4,5,2)(3,6,8,7)(9,13,14,15)(10,17,18,11)(12,22,20,23)(16,21,24,19)(25,26)(27,29)(28,32)(30,31)(33,35)(34,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=i,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8924645);