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 | #36211 | 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:
(1,2)(3,6)(4,7)(5,8)(9,10,15,14)(11,19,22,16)(12,23,21,17)(13,24,20,18)(28,36)(32,39)(35,40)(37,38);(1,3,5,7)(2,6,8,4)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,18)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3)(2,4)(5,7)(6,8)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,12,15,21)(10,17,14,23)(11,20,22,13)(16,18,19,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,17,19,23)(25,28,30,36)(26,32,34,39)(27,35,33,40)(29,37,31,38);(1,4,5,6)(2,7,8,3)(9,14,15,10)(11,19,22,16)(12,17,21,23)(13,24,20,18)(25,29,30,31)(26,33,34,27)(28,37,36,38)(32,40,39,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=ghi,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, 8365994);