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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,12)(11,16,19,20)(13,17,23,21)(14,22,24,18)(25,26,30,31)(27,33,36,32)(28,29,37,38)(34,39,40,35);(1,2,5,4)(3,7,8,6)(9,11)(10,16)(12,20)(13,22)(14,21)(15,19)(17,24)(18,23)(26,31)(28,37)(32,33)(34,40);(1,3,5,8)(2,6,4,7)(10,12)(11,19)(13,23)(14,24)(25,26)(27,32)(28,29)(30,31)(33,36)(34,35)(37,38)(39,40);(3,8)(6,7)(9,12,15,10)(11,20,19,16)(13,17,23,21)(14,22,24,18)(25,27,30,36)(26,32,31,33)(28,34,37,40)(29,35,38,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,17)(11,18)(12,21)(14,16)(15,23)(19,22)(20,24)(25,28)(26,29)(27,34)(30,37)(31,38)(32,35)(33,39)(36,40);(1,4,5,2)(3,6,8,7)(9,14,15,24)(10,18,12,22)(11,21,19,17)(13,16,23,20)(25,29)(26,28)(27,35)(30,38)(31,37)(32,34)(33,40)(36,39)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=hi,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8935612);