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 | |
4438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24);(1,3)(2,4)(5,8)(6,7)(10,11)(15,17)(16,18)(22,23)(25,26,30,31)(27,33,35,29)(28,32,37,38)(34,39,40,36);(1,3,5,8)(2,6,7,4)(13,21)(16,18)(19,24)(22,23)(26,31)(29,33)(32,38)(36,39);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27)(26,29)(28,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,37)(26,32,31,38)(27,34,35,40)(29,36,33,39);(1,3)(2,4)(5,8)(6,7)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29)(26,27)(28,36)(30,33)(31,35)(32,34)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=g,e^2=ghi,f^2=g,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, 8099637);