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 | #34547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(10,14)(12,20)(15,17)(18,24)(25,26)(27,35)(28,38)(29,34)(30,33)(31,32)(36,39)(37,40);(2,3)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(26,32)(28,35)(33,34)(39,40);(1,3,5,2)(4,7,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,27,31,38)(26,28,32,35)(29,36,30,37)(33,39,34,40);(1,4)(2,6)(3,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28)(26,27)(29,39)(30,40)(31,35)(32,38)(33,36)(34,37);(25,29,31,30)(26,33,32,34)(27,36,38,37)(28,39,35,40);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30,31,29)(26,34,32,33)(27,37,38,36)(28,40,35,39)Polycyclic presentation:
2^9|a^2=i,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015204);