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 | #46328 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(12,20)(16,21)(19,24)(22,23)(26,30)(28,35)(31,36)(34,39);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,21)(13,17)(14,18)(16,20)(19,22)(23,24)(27,33)(31,36)(34,39)(38,40);(9,11)(10,13)(12,19)(14,17)(15,18)(16,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(2,4)(6,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,24)(21,22)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,20)(10,16,18,21)(11,19,17,24)(13,22,15,23)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(1,4,5,2)(3,7,8,6)(9,13,14,15)(10,17,18,11)(12,22,20,23)(16,24,21,19)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8413226);