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 | #46604 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,13)(14,15)(17,22)(18,21)(19,23)(20,24);(2,3)(6,7)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,20)(23,24)(25,26,30,33)(27,34,37,31)(28,29,32,38)(35,40,39,36);(1,3,5,2)(4,7,8,6)(10,15)(12,22)(16,18)(19,24)(25,27,30,37)(26,31,33,34)(28,35,32,39)(29,36,38,40);(1,4,5,8)(2,6,3,7)(9,11)(10,16)(12,19)(13,23)(14,21)(15,18)(17,20)(22,24)(25,28)(26,29)(27,35)(30,32)(31,36)(33,38)(34,40)(37,39);(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,23)(21,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(9,13,14,17)(10,12,15,22)(11,20,21,23)(16,24,18,19)(25,29)(26,32)(27,36)(28,33)(30,38)(31,39)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8514626);