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 | #54925 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4283 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)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,39,37,36);(2,3)(6,7)(10,15)(11,18)(17,20)(19,24)(25,27,30,35)(26,31,33,28)(29,34,38,37)(32,39,40,36);(1,3,5,2)(4,7,8,6)(9,10,14,15)(11,16,18,13)(12,17,22,20)(19,23,24,21)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,38)(39,40);(1,4,5,8)(2,6,3,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,29,30,38)(26,32,33,40)(27,34,35,37)(28,36,31,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=g,d^2=gh,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8163823);