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 | |
4634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26,31,33)(27,34,29,32)(28,37,38,30)(35,36,39,40);(2,3)(6,7)(10,15)(12,17)(16,21)(20,23)(25,27,31,29)(26,32,33,34)(28,35,38,39)(30,36,37,40);(1,3,5,2)(4,7,8,6)(10,15)(11,18)(13,22)(20,23)(28,38)(30,37)(35,39)(36,40);(1,4,5,8)(2,6,3,7)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,28)(26,30)(27,35)(29,39)(31,38)(32,36)(33,37)(34,40);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29,31,27)(26,32,33,34)(28,35,38,39)(30,40,37,36);(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,38)(34,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=i,d^2=hi,e^2=ghi,f^2=h,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 subgroup of | #72219287 |
| maximal normal subgroup of | #72219281, #72219287 |
| maximal quotient of | #72219275 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8831953);