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 | |
4572 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)(9,10,14,16)(11,19,20,15)(12,17,18,13)(21,23,24,22);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,21)(13,23)(14,20)(16,19)(17,22)(18,24)(25,26,30,31)(27,32,36,38)(28,35,39,33)(29,34,40,37);(1,3,5,4)(2,7,6,8)(10,16)(11,20)(12,18)(22,23)(26,31)(29,40)(32,38)(33,35);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,21)(13,16)(14,18)(15,22)(19,23)(20,24)(25,27)(26,32)(28,37)(29,35)(30,36)(31,38)(33,40)(34,39);(9,13,14,17)(10,18,16,12)(11,22,20,23)(15,21,19,24)(25,28,30,39)(26,33,31,35)(27,34,36,37)(29,38,40,32);(1,4,5,3)(2,8,6,7)(9,11)(10,19)(12,21)(13,22)(14,20)(15,16)(17,23)(18,24)(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=hi,c^2=gh,d^2=gh,e^2=i,f^2=gh,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, 7702945);