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 | #46090 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(10,11)(12,13)(14,22)(23,24)(26,31)(28,36)(29,37)(32,38);(2,3)(6,7)(9,10,15,11)(12,17,13,16)(14,18,22,19)(20,24,21,23)(26,31)(27,35)(28,36)(34,40);(1,2,5,3)(4,6,8,7)(9,11,15,10)(12,17,13,16)(14,19,22,18)(20,24,21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(9,12,15,13)(10,16,11,17)(14,20,22,21)(18,23,19,24)(25,27,30,35)(26,32,31,38)(28,29,36,37)(33,34,39,40);(1,3,5,2)(4,7,8,6)(9,13,15,12)(10,17,11,16)(14,21,22,20)(18,24,19,23)(25,28,30,36)(26,33,31,39)(27,29,35,37)(32,34,38,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,29)(26,34)(27,36)(28,35)(30,37)(31,40)(32,39)(33,38)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017168);