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 | |
4471 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)(11,19)(12,21)(13,18)(14,15)(16,20)(17,22)(23,24);(2,3)(6,7)(10,15)(11,20)(17,21)(18,24)(25,26,30,31)(27,32,36,39)(28,33,37,29)(34,35,40,38);(1,3,5,2)(4,7,8,6)(10,15)(12,22)(16,19)(18,24)(26,31)(28,37)(32,39)(34,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,16)(12,18)(13,17)(14,20)(15,19)(21,23)(22,24)(25,27)(26,32)(28,34)(29,38)(30,36)(31,39)(33,35)(37,40);(9,12,14,22)(10,17,15,21)(11,18,20,24)(13,19,23,16)(25,28)(26,33)(27,34)(29,31)(30,37)(32,35)(36,40)(38,39);(9,13)(10,18)(11,21)(12,19)(14,23)(15,24)(16,22)(17,20)(25,29,30,33)(26,28,31,37)(27,35,36,38)(32,40,39,34)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8456764);