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 | |
4539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,24)(25,26)(27,35)(28,37)(29,34)(30,33)(31,32)(36,39)(38,40);(3,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27)(26,29)(28,36)(30,40)(31,35)(32,34)(33,39)(37,38);(1,2)(3,5)(4,6)(7,8)(11,18)(14,15)(19,24)(22,23)(25,28,32,30)(26,33,31,37)(27,36,34,40)(29,39,35,38);(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,29,32,35)(26,34,31,27)(28,38,30,39)(33,36,37,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,20)(13,17)(15,18)(16,21)(19,23)(22,24)(25,30,32,28)(26,33,31,37)(27,36,34,40)(29,38,35,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,31,32,26)(27,35,34,29)(28,37,30,33)(36,38,40,39)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=g,e^2=g,f^2=g,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, 9073754);