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 | |
4309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,19,24)(12,17,21,20)(13,22,23,18)(25,26,30,31)(27,33,36,32)(28,29,37,38)(34,39,40,35);(1,2)(3,4)(5,7)(6,8)(9,11,14,19)(10,15,16,24)(12,18,21,22)(13,20,23,17)(26,31)(28,37)(32,33)(34,40);(1,3)(2,6)(4,7)(5,8)(10,16)(11,19)(12,21)(13,23)(25,26)(27,32)(28,29)(30,31)(33,36)(34,35)(37,38)(39,40);(3,8)(4,6)(9,12)(10,17)(11,18)(13,15)(14,21)(16,20)(19,22)(23,24)(25,27,30,36)(26,32,31,33)(28,34,37,40)(29,35,38,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,22)(13,20)(14,19)(16,24)(17,23)(18,21)(25,28)(26,29)(27,34)(30,37)(31,38)(32,35)(33,39)(36,40);(1,2)(3,4)(5,7)(6,8)(9,13,14,23)(10,18,16,22)(11,20,19,17)(12,15,21,24)(25,29)(26,28)(27,35)(30,38)(31,37)(32,34)(33,40)(36,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8877730);