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 | #44909 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,12)(11,16,14,18)(13,20,22,17)(19,24,21,23)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,40)(38,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,18)(13,19)(14,15)(17,23)(20,24)(21,22)(25,27,30,36)(26,31,34,35)(28,37,38,29)(32,33,39,40);(1,3,5,4)(2,6,8,7)(10,12)(11,14)(17,20)(19,21)(27,36)(29,37)(31,35)(33,40);(1,3,5,4)(2,6,8,7)(9,12,15,10)(11,18,14,16)(13,17,22,20)(19,23,21,24)(25,28,30,38)(26,32,34,39)(27,37,36,29)(31,33,35,40);(1,4,5,3)(2,7,8,6)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,26)(27,31)(28,39)(29,33)(30,34)(32,38)(35,36)(37,40);(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,29,30,37)(26,33,34,40)(27,38,36,28)(31,32,35,39)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=gh,e^2=gi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7756014);