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 | |
4519 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)(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,34)(28,32)(29,37)(30,31)(33,39)(35,36)(38,40);(1,3,5,8)(2,4,6,7)(12,19)(16,22)(18,23)(20,24)(25,27,30,32)(26,28,31,34)(29,35,39,40)(33,36,37,38);(2,6)(3,8)(10,15)(11,17)(16,22)(18,23)(26,31)(28,34)(33,37)(36,38);(1,3)(2,7)(4,6)(5,8)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,38)(36,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28,30,34)(26,32,31,27)(29,38,39,36)(33,35,37,40);(1,4)(2,8)(3,6)(5,7)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29,30,39)(26,33,31,37)(27,35,32,40)(28,36,34,38)Polycyclic presentation:
2^9|b^2=gi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9948604);