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 | #52297 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,19)(12,22)(13,18)(14,15)(16,20)(17,23)(21,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(2,6)(3,7)(10,15)(11,20)(17,22)(18,21)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,3)(2,4)(5,7)(6,8)(10,15)(11,20)(17,22)(18,21)(28,37)(32,36)(34,40)(38,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,24)(18,23)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,16,24,19)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,38)(37,40);(9,13,14,24)(10,18,15,21)(11,17,20,22)(12,19,23,16)(25,29,30,26)(27,33,35,31)(28,32,37,36)(34,38,40,39)Polycyclic presentation:
2^9|a^2=gh,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8631993);