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 | #53424 | 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) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(25,26,32,35)(27,33,30,31)(28,38,29,34)(36,39,37,40);(1,3,5,8)(2,4,6,7)(10,15)(11,18)(17,20)(19,24)(25,27,32,30)(26,33,35,31)(28,37,29,36)(34,39,38,40);(1,2)(3,4)(5,6)(7,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,28,32,29)(26,34,35,38)(27,36,30,37)(31,39,33,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,29,32,28)(26,34,35,38)(27,37,30,36)(31,39,33,40);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,30,32,27)(26,31,35,33)(28,37,29,36)(34,39,38,40);(1,4,5,7)(2,3,6,8)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,31)(26,27)(28,39)(29,40)(30,35)(32,33)(34,36)(37,38)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gh,d^2=gh,e^2=i,f^2=g,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, 8353140);