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 | #42927 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,13)(12,20)(14,15)(16,18)(17,21)(19,24)(22,23)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,37,39,36);(1,2,5,6)(3,7,8,4)(10,15)(11,18)(12,21)(23,24)(25,27,30,35)(26,28,33,31)(29,34,38,39)(32,36,40,37);(2,6)(4,7)(10,15)(13,16)(17,20)(23,24)(25,28,30,31)(26,27,33,35)(29,37,38,36)(32,39,40,34);(1,3,5,8)(2,7,6,4)(9,11)(10,16)(12,22)(13,15)(14,18)(17,23)(19,21)(20,24)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,21)(10,17,15,20)(11,19,18,22)(13,23,16,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,37)(36,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,16)(10,11,15,18)(12,23,21,24)(17,19,20,22)(25,29,30,38)(26,32,33,40)(27,34,35,39)(28,36,31,37)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8265924);