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 | #49590 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,14,19,22)(12,17,21,23)(13,18,20,24)(25,26)(27,33)(28,36)(29,31)(30,35)(32,38)(34,39)(37,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,19)(10,14,16,22)(12,20,21,13)(17,24,23,18)(25,27,29,35)(26,30,31,33)(28,37,38,34)(32,40,36,39);(2,5)(6,7)(10,16)(11,19)(13,20)(17,23)(26,31)(30,33)(32,36)(39,40);(2,5)(6,7)(9,12)(10,17)(11,13)(14,18)(15,21)(16,23)(19,20)(22,24)(25,28,29,38)(26,32,31,36)(27,34,35,37)(30,39,33,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,12)(14,17)(15,20)(16,24)(19,21)(22,23)(25,26)(27,30)(28,32)(29,31)(33,35)(34,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,14)(10,11)(12,18)(13,17)(15,22)(16,19)(20,23)(21,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,39)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227835);