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 | #42616 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,21)(12,18)(13,15)(14,24)(16,20)(17,23)(19,22)(28,36)(33,37)(34,40)(38,39);(3,8)(6,7)(9,11,16,23)(10,17,20,21)(12,22,13,14)(15,19,18,24)(25,26,30,29)(27,32,35,31)(28,33,36,37)(34,39,40,38);(1,3)(2,6)(4,7)(5,8)(9,12,16,13)(10,18,20,15)(11,22,23,14)(17,24,21,19)(25,27)(26,31)(28,34)(29,32)(30,35)(33,38)(36,40)(37,39);(1,3)(2,7)(4,6)(5,8)(9,13,16,12)(10,18,20,15)(11,14,23,22)(17,24,21,19)(25,27)(26,32)(28,34)(29,31)(30,35)(33,39)(36,40)(37,38);(9,14)(10,19)(11,12)(13,23)(15,21)(16,22)(17,18)(20,24)(25,28,30,36)(26,33,29,37)(27,34,35,40)(31,38,32,39);(1,4,5,2)(3,7,8,6)(9,15,16,18)(10,13,20,12)(11,19,23,24)(14,21,22,17)(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=ghi,c^2=g,d^2=g,e^2=hi,f^2=gi,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, 8368440);