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 | |
4397 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)(11,20,23,16)(12,18,24,14)(13,21,22,19);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,21)(13,18)(14,22)(15,23)(17,20)(19,24)(25,26,30,31)(27,32,36,38)(28,35,39,33)(29,34,40,37);(1,3,5,4)(2,7,6,8)(10,17)(12,24)(16,20)(19,21)(26,31)(29,40)(32,38)(33,35);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,21)(13,16)(14,17)(15,24)(19,23)(20,22)(25,27)(26,32)(28,37)(29,35)(30,36)(31,38)(33,40)(34,39);(1,4,5,3)(2,8,6,7)(9,13,15,22)(10,19,17,21)(11,18,23,14)(12,16,24,20)(25,28,30,39)(26,33,31,35)(27,34,36,37)(29,38,40,32);(9,14,15,18)(10,12,17,24)(11,22,23,13)(16,21,20,19)(25,29)(26,34)(27,35)(28,38)(30,40)(31,37)(32,39)(33,36)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7702931);