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 | |
4525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,31,29)(27,33,35,28)(30,37,38,32)(34,36,40,39);(3,8)(5,6)(12,19)(15,17)(20,24)(22,23)(25,27,31,35)(26,28,29,33)(30,34,38,40)(32,36,37,39);(1,3,4,8)(2,5,7,6)(10,11)(15,17)(16,18)(22,23)(26,29)(28,33)(32,37)(36,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,31,26)(27,28,35,33)(30,32,38,37)(34,39,40,36);(1,2)(3,5)(4,7)(6,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,32)(27,34)(28,36)(29,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9952832);