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 | |
4565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26,31,35)(27,36,37,32)(28,29,33,39)(30,34,38,40);(1,2)(3,5)(4,6)(7,8)(9,10,14,15)(11,12,16,20)(13,21,22,17)(18,24,23,19)(25,27)(26,32)(28,38)(29,34)(30,33)(31,37)(35,36)(39,40);(10,15)(11,16)(13,22)(19,24)(27,37)(30,38)(32,36)(34,40);(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,24)(22,23)(25,28)(26,29)(27,38)(30,37)(31,33)(32,34)(35,39)(36,40);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,21)(22,24)(25,29)(26,33)(27,34)(28,35)(30,32)(31,39)(36,38)(37,40);(1,3,4,7)(2,5,6,8)(9,13,14,22)(10,17,15,21)(11,18,16,23)(12,19,20,24)(25,30,31,38)(26,34,35,40)(27,28,37,33)(29,32,39,36)Polycyclic presentation:
2^9|a^2=gh,b^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7719274);