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 | #51941 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10)(11,14)(12,20)(13,15)(16,18)(17,21)(19,23)(22,24)(25,26,32,36)(27,37,31,33)(28,39,38,34)(29,40,35,30);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,19)(13,18)(15,16)(17,22)(20,23)(21,24)(25,27,32,31)(26,33,36,37)(28,35,38,29)(30,39,40,34);(1,2,4,6)(3,5,7,8)(10,15)(11,18)(17,20)(19,24)(25,28)(26,34)(27,29)(30,37)(31,35)(32,38)(33,40)(36,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,23)(20,21)(22,24)(25,29,32,35)(26,30,36,40)(27,38,31,28)(33,34,37,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,18)(14,15)(17,23)(20,22)(21,24)(25,30)(26,35)(27,39)(28,37)(29,36)(31,34)(32,40)(33,38);(1,3)(2,5)(4,7)(6,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,31,32,27)(26,33,36,37)(28,35,38,29)(30,34,40,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8033924);