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 | #59550 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,12)(11,15,17,18)(13,16,22,20)(19,23,21,24)(25,26)(27,28)(29,34)(30,35)(31,32)(33,40)(36,38)(37,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,18)(13,21)(14,17)(16,24)(19,22)(20,23)(26,32)(27,33)(34,35)(36,37);(2,4)(6,7)(11,17)(15,18)(19,21)(23,24)(25,27,31,33)(26,28,32,40)(29,36,30,37)(34,38,35,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,18,17,15)(13,20,22,16)(19,24,21,23)(25,28)(26,33)(27,32)(29,38)(30,39)(31,40)(34,37)(35,36);(1,3)(2,6)(4,7)(5,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,29,31,30)(26,34,32,35)(27,36,33,37)(28,38,40,39);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,24)(21,22)(25,30,31,29)(26,35,32,34)(27,37,33,36)(28,39,40,38)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8118894);