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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4520 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)(11,14)(12,15)(13,18)(16,21)(17,20)(19,23)(22,24)(25,26)(27,32)(28,30)(29,33)(31,35)(34,40)(36,38)(37,39);(1,2)(3,8)(4,6)(5,7)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27)(26,31)(28,32)(29,34)(30,35)(33,39)(36,40)(37,38);(1,3,5,6)(2,4,7,8)(10,11)(15,17)(16,18)(22,23);(1,4)(2,6)(3,7)(5,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,30,26)(27,32,35,31)(29,36,38,33)(34,40,37,39);(1,2)(3,4)(5,7)(6,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,27)(26,32)(28,31)(29,37)(30,35)(33,39)(34,38)(36,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10009825);