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 | #54403 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(26,32)(27,28)(34,38)(35,37);(1,3,5,7)(2,4,8,6)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,26,31,32)(27,33,28,36)(29,34,39,38)(30,35,40,37);(1,3)(2,6)(4,8)(5,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,31,28)(26,33,32,36)(29,35,39,37)(30,34,40,38);(1,3)(2,4)(5,7)(6,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,28,31,27)(26,33,32,36)(29,37,39,35)(30,34,40,38);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,29,31,39)(26,34,32,38)(27,35,28,37)(30,33,40,36);(1,4)(2,7)(3,8)(5,6)(9,10)(11,15)(12,16)(13,22)(14,17)(18,21)(19,23)(20,24)(25,30,31,40)(26,35,32,37)(27,34,28,38)(29,33,39,36)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8367913);