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 | #40598 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,15)(11,14,17,22)(12,16,19,23)(18,21,20,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,2,5,4)(3,7,8,6)(9,11,13,17)(10,14,15,22)(12,18,19,20)(16,21,23,24)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(1,3,5,8)(2,6,4,7)(10,15)(11,17)(12,19)(21,24)(25,27,30,28)(26,32,34,31)(29,36,37,35)(33,38,40,39);(1,3,5,8)(2,7,4,6)(9,12)(10,16)(11,18)(13,19)(14,21)(15,23)(17,20)(22,24)(25,28,30,27)(26,32,34,31)(29,35,37,36)(33,38,40,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,20)(13,17)(15,22)(16,24)(18,19)(21,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,4,5,2)(3,6,8,7)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,23)(20,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=ghi,d^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9515172);