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 | |
4374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(25,26)(27,35)(28,33)(29,31)(30,34)(32,37)(36,38)(39,40);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19)(25,27,31,37)(26,32,29,35)(28,30,36,40)(33,39,38,34);(10,15)(13,23)(16,24)(17,20)(25,26,31,29)(27,35,37,32)(28,38,36,33)(30,39,40,34);(9,11)(10,16)(12,18)(13,20)(14,21)(15,24)(17,23)(19,22)(25,28,31,36)(26,33,29,38)(27,30,37,40)(32,39,35,34);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,29,31,26)(27,35,37,32)(28,33,36,38)(30,39,40,34);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,30)(26,34)(27,36)(28,37)(29,39)(31,40)(32,33)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7675272);