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 | |
4668 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)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(25,26)(27,32)(28,37)(29,34)(30,35)(31,38)(33,39)(36,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(11,17)(15,19)(18,23)(22,24)(25,27,30,38)(26,31,35,32)(28,36,39,34)(29,33,40,37);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,27)(26,32)(28,34)(29,37)(30,38)(31,35)(33,40)(36,39);(1,3,5,4)(2,7,6,8)(9,13,14,10)(11,19,17,15)(12,16,20,21)(18,22,23,24)(25,28)(26,33)(27,36)(29,32)(30,39)(31,40)(34,38)(35,37);(1,4,5,3)(2,8,6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,29)(26,34)(27,37)(28,32)(30,40)(31,39)(33,38)(35,36)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7681121);