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 | #41030 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4403 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)(11,17)(12,14)(13,23)(15,18)(16,22)(19,24)(20,21)(25,26,29,33)(27,31,35,30)(28,36,38,32)(34,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,27,29,35)(26,30,33,31)(28,34,38,37)(32,39,36,40);(2,5)(3,7)(9,12,15,13)(10,14,18,23)(11,19,22,20)(16,21,17,24)(25,27,29,35)(26,30,33,31)(28,37,38,34)(32,40,36,39);(3,7)(6,8)(9,13,15,12)(10,14,18,23)(11,20,22,19)(16,21,17,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,20)(13,19)(14,21)(15,22)(16,18)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,14)(10,12)(11,21)(13,18)(15,23)(16,19)(17,20)(22,24)(25,28,29,38)(26,32,33,36)(27,34,35,37)(30,39,31,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8843843);