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 | |
4528 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)(3,5)(4,7)(6,8)(9,10,14,12)(11,17,18,15)(13,16,21,20)(19,23,24,22)(25,26,31,34)(27,30,33,40)(28,29,32,39)(35,36,37,38);(2,3)(6,7)(10,12)(11,18)(16,20)(19,24)(25,27)(26,30)(28,37)(29,38)(31,33)(32,35)(34,40)(36,39);(1,2,5,3)(4,6,8,7)(13,21)(16,20)(19,24)(22,23)(27,33)(30,40)(35,37)(36,38);(1,3,5,2)(4,7,8,6)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,28)(26,29)(27,35)(30,36)(31,32)(33,37)(34,39)(38,40);(1,4,5,8)(2,6,3,7)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,29)(26,32)(27,36)(28,34)(30,37)(31,39)(33,38)(35,40);(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,30,31,40)(26,33,34,27)(28,36,32,38)(29,37,39,35)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306693);