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 | |
4446 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)(3,4)(5,6)(7,8)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(26,31)(27,33)(32,39)(34,35);(2,6)(4,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,2,5,6)(3,7,8,4)(11,17)(12,20)(15,19)(16,21)(27,33)(29,38)(34,35)(36,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,20)(10,16,13,21)(11,18,17,23)(15,22,19,24)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(1,4,5,7)(2,3,6,8)(9,13,14,10)(11,19,17,15)(12,16,20,21)(18,22,23,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,39)(33,35)(38,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=h,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219283, #72219285 |
| maximal quotient of | #72219271 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8829724);