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 | #2433 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(13,21)(16,18)(19,24)(22,23)(25,26,30,32)(27,33,36,31)(28,37,38,29)(34,35,40,39);(1,3)(2,5)(4,8)(6,7)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27)(26,31)(28,34)(29,35)(30,36)(32,33)(37,39)(38,40);(1,3)(2,5)(4,8)(6,7)(10,11)(15,17)(16,18)(22,23)(25,26,30,32)(27,31,36,33)(28,37,38,29)(34,39,40,35);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28,30,38)(26,29,32,37)(27,34,36,40)(31,35,33,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29)(26,28)(27,35)(30,37)(31,34)(32,38)(33,40)(36,39);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,33)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8074590);