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 |
| prime radical | #59401 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,17)(12,15)(13,18)(14,16)(19,24)(20,22)(21,23)(25,26)(27,35)(28,31)(29,38)(30,34)(32,36)(33,40)(37,39);(1,2)(3,6)(4,7)(5,8)(9,11,14,12)(10,15,16,17)(13,19,21,20)(18,22,23,24)(25,27)(26,32)(28,35)(29,37)(30,40)(31,36)(33,39)(34,38);(1,3,5,4)(2,6,8,7)(10,16)(15,17)(18,23)(22,24)(26,28)(27,36)(29,40)(34,39);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,28,31,26)(27,35,36,32)(29,38,40,33)(30,39,37,34);(1,3,5,4)(2,6,8,7)(9,12,14,11)(10,17,16,15)(13,20,21,19)(18,24,23,22)(25,29,31,40)(26,33,28,38)(27,37,36,30)(32,39,35,34);(1,4,5,3)(2,7,8,6)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,30,31,37)(26,34,28,39)(27,29,36,40)(32,33,35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7888417);