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 | |
4382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,19)(12,13)(14,16)(15,17)(18,24)(20,21)(22,23)(25,26)(27,35)(28,39)(29,34)(30,32)(31,38)(33,37)(36,40);(1,3,5,8)(2,6,4,7)(9,11,15,14)(10,16,17,19)(12,20,18,22)(13,23,24,21)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(1,2,5,4)(3,6,8,7)(10,17)(11,14)(12,18)(21,23)(26,32)(31,35)(33,39)(34,40);(1,2,5,4)(3,6,8,7)(9,12)(10,13)(11,20)(14,22)(15,18)(16,23)(17,24)(19,21)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,21)(12,17)(14,23)(15,24)(16,20)(19,22)(25,28,30,37)(26,33,32,39)(27,36,38,29)(31,40,35,34);(9,14,15,11)(10,16,17,19)(12,22,18,20)(13,23,24,21)(25,29,30,36)(26,34,32,40)(27,37,38,28)(31,39,35,33)Polycyclic presentation:
2^9|a^2=gh,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, 7702932);