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 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,17)(11,15,20,23)(12,13,21,16)(18,19,24,22)(25,26,30,33)(27,34,29,31)(28,32,37,40)(35,39,36,38);(1,2,5,4)(3,6,8,7)(9,11,14,20)(10,15,17,23)(12,18,21,24)(13,19,16,22)(27,29)(31,34)(35,36)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,21)(10,16,17,13)(11,18,20,24)(15,22,23,19)(25,27,30,29)(26,31,33,34)(28,36,37,35)(32,39,40,38);(1,3,5,8)(2,6,4,7)(9,12)(10,13)(11,18)(14,21)(15,19)(16,17)(20,24)(22,23)(25,28,30,37)(26,32,33,40)(27,35,29,36)(31,38,34,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,29,30,27)(26,31,33,34)(28,35,37,36)(32,39,40,38);(1,4,5,2)(3,7,8,6)(9,13)(10,12)(11,19)(14,16)(15,18)(17,21)(20,22)(23,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10078827);