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 | #51905 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10,14,15)(11,18,17,13)(12,16,20,23)(19,24,21,22);(1,2,5,8)(3,6,4,7)(10,15)(11,17)(16,23)(19,21)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(1,3,5,4)(2,6,8,7)(10,15)(12,20)(13,18)(19,21)(26,30)(28,37)(31,33)(34,38);(9,11,14,17)(10,13,15,18)(12,21,20,19)(16,24,23,22)(25,27)(26,31)(28,38)(29,35)(30,33)(32,40)(34,37)(36,39);(1,2)(3,6)(4,7)(5,8)(9,12,14,20)(10,16,15,23)(11,19,17,21)(13,22,18,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38);(1,4,5,3)(2,7,8,6)(9,13)(10,17)(11,15)(12,22)(14,18)(16,21)(19,23)(20,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,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,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948126);