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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4540 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)(17,21)(18,24)(19,22)(20,23)(25,26,30,33)(27,29,32,34)(28,31,39,37)(35,38,36,40);(2,3)(6,7)(10,15)(12,21)(16,23)(18,22)(25,27)(26,29)(28,36)(30,32)(31,40)(33,34)(35,39)(37,38);(1,3,5,2)(4,7,8,6)(10,15)(13,17)(16,23)(19,24)(27,32)(29,34)(35,36)(38,40);(1,4,5,8)(2,6,3,7)(9,11,14,20)(10,16,15,23)(12,22,21,18)(13,19,17,24)(25,26)(27,34)(28,37)(29,32)(30,33)(31,39)(35,38)(36,40);(9,12)(10,17)(11,18)(13,15)(14,21)(16,19)(20,22)(23,24)(25,28)(26,31)(27,35)(29,38)(30,39)(32,36)(33,37)(34,40);(9,13,14,17)(10,12,15,21)(11,19,20,24)(16,22,23,18)(25,29,30,34)(26,32,33,27)(28,38,39,40)(31,36,37,35)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325568);