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 | |
4412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(13,21)(17,22)(18,23)(20,24)(25,26)(27,28)(29,37)(30,34)(31,32)(33,39)(35,40)(36,38);(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,27,31,30)(26,28,32,34)(29,35,39,38)(33,36,37,40);(1,3,5,4)(2,7,6,8)(10,15)(12,19)(17,22)(20,24)(26,32)(28,34)(33,37)(36,40);(9,11,14,16)(10,12,15,19)(13,18,21,23)(17,20,22,24)(25,28,31,34)(26,27,32,30)(29,36,39,40)(33,35,37,38);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,29,31,39)(26,33,32,37)(27,35,30,38)(28,36,34,40);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,20,19,24)(25,30,31,27)(26,34,32,28)(29,38,39,35)(33,40,37,36)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=hi,e^2=gh,f^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7628912);