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 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,12)(11,15,17,19)(13,20,21,16)(18,24,23,22)(26,31)(27,34)(33,39)(35,37);(1,2,5,4)(3,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,39)(37,38);(1,3,5,8)(2,6,4,7)(10,12)(11,17)(13,21)(22,24)(27,34)(28,36)(29,32)(33,39);(3,8)(6,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27,30,34)(26,32,31,29)(28,37,36,35)(33,38,39,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28,30,36)(26,33,31,39)(27,35,34,37)(29,38,32,40);(1,4,5,2)(3,6,8,7)(9,13,14,21)(10,16,12,20)(11,18,17,23)(15,22,19,24)(25,29,30,32)(26,34,31,27)(28,38,36,40)(33,37,39,35)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=hi,d^2=g,e^2=gh,f^2=gi,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, 8935736);