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 | #47859 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,17)(13,21)(14,15)(18,23)(19,20)(22,24)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(2,3)(6,7)(10,15)(12,16)(18,21)(20,22)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,3,5,2)(4,7,8,6)(10,15)(11,17)(18,21)(19,24)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(1,4,5,8)(2,6,3,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,22)(20,21)(23,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,21,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(9,13,14,23)(10,18,15,21)(11,19,17,24)(12,20,16,22)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113673);