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 | #38364 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,16)(11,15,19,24)(12,17,21,22)(13,20,23,18)(25,26,32,36)(27,37,39,33)(28,31,29,34)(30,35,38,40);(9,11)(10,15)(12,20)(13,22)(14,19)(16,24)(17,23)(18,21)(25,27)(26,33)(28,35)(29,40)(30,31)(32,39)(34,38)(36,37);(1,2)(3,6)(4,5)(7,8)(10,16)(11,19)(17,22)(18,20)(25,28,32,29)(26,34,36,31)(27,35,39,40)(30,33,38,37);(3,7)(6,8)(9,12)(10,17)(11,18)(13,15)(14,21)(16,22)(19,20)(23,24)(25,29,32,28)(26,31,36,34)(27,35,39,40)(30,33,38,37);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,24)(20,21)(22,23)(25,30,32,38)(26,35,36,40)(27,31,39,34)(28,37,29,33);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,17)(12,15)(14,23)(16,20)(19,22)(21,24)(25,31)(26,28)(27,38)(29,36)(30,39)(32,34)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=gi,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231698);