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 | |
4521 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(4,8)(9,10)(11,15)(12,16)(13,22)(14,18)(17,23)(19,20)(21,24)(25,26)(27,28)(29,37)(30,34)(31,32)(33,38)(35,39)(36,40);(2,3)(4,8)(9,11)(10,15)(12,13)(14,19)(16,22)(17,24)(18,20)(21,23)(25,26,30,34)(27,31,32,28)(29,37,38,33)(35,36,40,39);(1,2,5,3)(4,6,8,7)(11,19)(12,21)(15,20)(16,24)(25,27,30,32)(26,31,34,28)(29,35,38,40)(33,39,37,36);(2,3)(6,7)(9,12,14,21)(10,16,18,24)(11,13,19,23)(15,22,20,17)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(1,3,5,2)(4,7,8,6)(9,13)(10,17)(11,12)(14,23)(15,24)(16,20)(18,22)(19,21)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(1,4,5,8)(2,6,3,7)(9,10)(11,20)(12,16)(13,17)(14,18)(15,19)(21,24)(22,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746314);