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 | #37666 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10)(11,19)(12,23)(13,24)(14,18)(15,22)(16,21)(17,20)(25,26,30,34)(27,32,35,31)(28,29,33,39)(36,38,40,37);(1,2,4,6)(3,5,7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,19)(23,24)(27,35)(31,32)(36,40)(37,38);(1,3)(2,5)(4,7)(6,8)(25,27)(26,31)(28,36)(29,37)(30,35)(32,34)(33,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,21)(10,16,18,23)(11,20,22,13)(15,17,19,24)(25,27)(26,32)(28,36)(29,38)(30,35)(31,34)(33,40)(37,39);(1,3)(2,5)(4,7)(6,8)(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,16,19,23)(25,28,30,33)(26,29,34,39)(27,36,35,40)(31,37,32,38);(9,10)(11,15)(12,23)(13,17)(14,18)(16,21)(19,22)(20,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=g,e^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, 8352797);