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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4448 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)(3,5)(4,7)(6,8)(13,21)(16,18)(19,24)(22,23)(26,29)(27,33)(32,37)(34,36);(1,2,5,3)(4,6,8,7)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,26,30,29)(27,31,33,35)(28,32,38,37)(34,39,36,40);(1,2,5,3)(4,6,8,7)(10,11)(15,17)(16,18)(22,23)(27,33)(31,35)(34,36)(39,40);(2,3)(6,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27)(26,31)(28,36)(29,35)(30,33)(32,40)(34,38)(37,39);(1,3,5,2)(4,7,8,6)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,38)(26,32,29,37)(27,34,33,36)(31,39,35,40);(1,4,5,8)(2,6,3,7)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,26)(27,35,33,31)(28,37,38,32)(34,40,36,39)Polycyclic presentation:
2^9|b^2=gi,c^2=h,d^2=i,e^2=gi,f^2=g,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, 8074627);