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 |
| prime radical | #45018 | C2 |
| 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) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,18)(13,17)(14,16)(19,22)(20,21)(23,24)(25,26,31,33)(27,36,30,32)(28,34,38,40)(29,37,39,35);(2,3)(6,7)(9,11,14,18)(10,15,16,12)(13,19,23,21)(17,22,24,20)(25,27,31,30)(26,32,33,36)(28,35,38,37)(29,34,39,40);(1,2,5,3)(4,6,8,7)(10,16)(11,18)(17,24)(19,21)(26,33)(28,38)(32,36)(35,37);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,21)(14,18)(15,16)(17,22)(19,23)(20,24)(25,28,31,38)(26,34,33,40)(27,37,30,35)(29,36,39,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,22)(14,15)(16,18)(17,21)(19,24)(20,23)(25,29)(26,35)(27,34)(28,36)(30,40)(31,39)(32,38)(33,37);(9,13,14,23)(10,17,16,24)(11,19,18,21)(12,20,15,22)(25,30,31,27)(26,32,33,36)(28,37,38,35)(29,34,39,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657070);