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 | |
4283 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)(9,10,15,17)(11,16,14,18)(12,13,21,24)(19,20,22,23)(25,26,31,33)(27,29,34,32)(28,37,38,30)(35,36,40,39);(1,2,5,6)(3,7,4,8)(9,11,15,14)(10,16,17,18)(12,19,21,22)(13,20,24,23)(25,27,31,34)(26,32,33,29)(28,35,38,40)(30,36,37,39);(10,17)(12,21)(16,18)(19,22)(26,33)(27,34)(28,38)(36,39);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,19)(14,22)(15,21)(16,20)(17,24)(18,23)(25,28)(26,30)(27,35)(29,39)(31,38)(32,36)(33,37)(34,40);(1,4,5,3)(2,8,6,7)(9,13)(10,12)(11,20)(14,23)(15,24)(16,19)(17,21)(18,22)(25,29)(26,27)(28,39)(30,35)(31,32)(33,34)(36,38)(37,40);(9,14,15,11)(10,18,17,16)(12,22,21,19)(13,23,24,20)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,38)(34,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=gi,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7868998);