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 | |
4488 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)(3,5)(4,6)(7,8)(9,10)(11,18)(12,15)(13,20)(14,17)(16,22)(19,21)(23,24)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(2,3)(6,7)(10,12)(11,17)(16,20)(19,23)(28,36)(32,38)(35,39)(37,40);(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,24,23,21)(25,28)(26,32)(27,35)(29,37)(30,38)(31,36)(33,39)(34,40);(9,13,15,22)(10,16,12,20)(11,19,17,23)(14,21,18,24)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(1,4)(2,6)(3,7)(5,8)(9,14,15,18)(10,17,12,11)(13,21,22,24)(16,23,20,19)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)Polycyclic presentation:
2^9|a^2=hi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8087567);