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 | #36724 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,20)(11,16,22,24)(12,13,18,17)(14,23,21,19)(26,32)(27,28)(30,40)(35,38);(1,2)(3,7)(4,5)(6,8)(9,11,15,22)(10,16,20,24)(12,21,18,14)(13,19,17,23)(25,26)(27,33)(28,37)(29,30)(31,32)(34,40)(35,39)(36,38);(2,5)(6,7)(9,12)(10,17)(11,21)(13,20)(14,22)(15,18)(16,23)(19,24)(25,27,31,28)(26,33,32,37)(29,38,34,35)(30,36,40,39);(2,5)(6,7)(9,12)(10,13)(11,21)(14,22)(15,18)(16,19)(17,20)(23,24)(25,28,31,27)(26,33,32,37)(29,38,34,35)(30,39,40,36);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,19)(12,20)(14,16)(15,17)(21,24)(22,23)(25,29,31,34)(26,30,32,40)(27,35,28,38)(33,39,37,36);(1,3)(2,6)(4,8)(5,7)(9,14,15,21)(10,19,20,23)(11,12,22,18)(13,16,17,24)(25,30)(26,34)(27,36)(28,39)(29,32)(31,40)(33,35)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=gh,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, 10229905);