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 | |
4478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,17)(11,16,20,22)(12,13,21,18)(14,19,23,24)(25,26,31,35)(27,36,39,32)(28,33,40,30)(29,37,38,34);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,14)(13,19)(15,20)(17,22)(18,24)(21,23)(25,27)(26,32)(28,37)(29,33)(30,38)(31,39)(34,40)(35,36);(10,17)(11,20)(12,21)(19,24)(25,28,31,40)(26,30,35,33)(27,37,39,34)(29,32,38,36);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,14)(13,17)(15,21)(16,24)(19,22)(20,23)(25,28)(26,33)(27,37)(29,32)(30,35)(31,40)(34,39)(36,38);(1,4,5,3)(2,8,6,7)(9,13,15,18)(10,12,17,21)(11,19,20,24)(14,22,23,16)(25,29,31,38)(26,34,35,37)(27,33,39,30)(28,32,40,36);(9,14)(10,19)(11,12)(13,16)(15,23)(17,24)(18,22)(20,21)(25,30,31,33)(26,28,35,40)(27,38,39,29)(32,37,36,34)Polycyclic presentation:
2^9|a^2=h,c^2=hi,d^2=ghi,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, 7813198);