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 | #48050 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26)(27,30)(28,29)(31,39)(32,34)(33,40)(35,36)(37,38);(1,2,5,7)(3,8,4,6)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27,32,29)(26,30,34,28)(31,35,40,37)(33,38,39,36);(1,3,5,4)(2,6,7,8)(10,15)(11,18)(16,21)(19,24)(25,28,32,30)(26,27,34,29)(31,36,40,38)(33,35,39,37);(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,29,32,27)(26,28,34,30)(31,37,40,35)(33,36,39,38);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,30,32,28)(26,27,34,29)(31,36,40,38)(33,37,39,35);(1,4,5,3)(2,6,7,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,31)(26,33)(27,35)(28,36)(29,37)(30,38)(32,40)(34,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9796713);