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 | |
4479 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)(9,10,15,12)(11,18,16,13)(14,17,23,20)(19,22,24,21)(25,26)(27,37)(28,34)(29,40)(30,36)(31,33)(32,39)(35,38);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,22)(23,24)(25,27)(26,32)(28,35)(29,34)(30,38)(31,39)(33,37)(36,40);(1,2,5,4)(3,6,8,7)(10,12)(11,16)(17,20)(19,24)(26,33)(28,36)(32,37)(35,40);(9,12,15,10)(11,18,16,13)(14,20,23,17)(19,22,24,21)(25,28,31,36)(26,34,33,30)(27,35,39,40)(29,37,38,32);(1,2,5,4)(3,6,8,7)(9,13,15,18)(10,16,12,11)(14,22,23,21)(17,19,20,24)(25,29,31,38)(26,35,33,40)(27,34,39,30)(28,37,36,32);(1,4,5,2)(3,7,8,6)(9,14,15,23)(10,17,12,20)(11,19,16,24)(13,21,18,22)(25,30,31,34)(26,36,33,28)(27,38,39,29)(32,40,37,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7759972);