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 | |
4364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,39)(38,40);(1,2)(3,7)(4,5)(6,8)(10,12)(15,17)(16,20)(22,23)(27,28)(29,32)(35,36)(37,39);(2,5)(6,7)(13,21)(16,20)(18,24)(22,23)(25,27,31,28)(26,32,34,29)(30,36,38,35)(33,37,40,39);(3,8)(6,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28,31,27)(26,32,34,29)(30,36,38,35)(33,39,40,37);(1,3,4,8)(2,6,5,7)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,29,31,32)(26,28,34,27)(30,37,38,39)(33,36,40,35);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,38)(26,33,34,40)(27,35,28,36)(29,37,32,39)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8269623);