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 | #57186 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,18)(11,15,19,22)(12,16,21,24)(13,17,23,20)(25,26)(27,33)(28,38)(29,35)(30,34)(31,32)(36,40)(37,39);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(25,27,32,29)(26,33,31,35)(28,37,30,36)(34,40,38,39);(1,2)(3,4)(5,7)(6,8)(11,19)(12,21)(15,22)(16,24)(25,28,32,30)(26,34,31,38)(27,36,29,37)(33,39,35,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,20)(13,15)(14,21)(17,19)(18,24)(22,23)(25,29,32,27)(26,35,31,33)(28,36,30,37)(34,39,38,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,16)(12,22)(14,23)(15,21)(18,20)(19,24)(25,30,32,28)(26,34,31,38)(27,37,29,36)(33,39,35,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(25,31,32,26)(27,35,29,33)(28,34,30,38)(36,39,37,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815553);