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 | #37346 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4393 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)(3,8)(4,6)(5,7)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,22,20,24)(28,36)(33,37)(34,40)(38,39);(1,3,5,4)(2,6,7,8)(10,16)(11,12)(18,21)(19,20)(25,26,30,29)(27,32,35,31)(28,33,36,37)(34,39,40,38);(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,22,21,24)(25,27)(26,31)(28,34)(29,32)(30,35)(33,38)(36,40)(37,39);(2,7)(6,8)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22)(25,27)(26,32)(28,34)(29,31)(30,35)(33,39)(36,40)(37,38);(1,4,5,3)(2,8,7,6)(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24)(25,28,30,36)(26,33,29,37)(27,34,35,40)(31,38,32,39);(1,2)(3,6)(4,8)(5,7)(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,19,21,20)(25,29,30,26)(27,32,35,31)(28,37,36,33)(34,39,40,38)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=h,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8368787);