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 | #42593 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(25,26)(27,38)(28,35)(29,34)(30,31)(32,37)(33,39)(36,40);(1,2,5,8)(3,6,4,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,27,32,39)(26,33,37,38)(28,30,29,40)(31,35,36,34);(1,3,5,4)(2,6,8,7)(10,15)(13,23)(16,21)(17,24)(25,28,32,29)(26,34,37,35)(27,30,39,40)(31,38,36,33);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,18)(13,17)(14,20)(15,21)(19,22)(23,24)(25,29,32,28)(26,35,37,34)(27,30,39,40)(31,38,36,33);(1,4,5,3)(2,7,8,6)(9,12)(10,17)(11,19)(13,21)(14,22)(15,24)(16,23)(18,20)(25,30)(26,36)(27,29)(28,39)(31,37)(32,40)(33,34)(35,38);(9,13,14,23)(10,18,15,19)(11,17,20,24)(12,21,22,16)(25,31,32,36)(26,30,37,40)(27,35,39,34)(28,33,29,38)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8229570);