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 | #37785 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,17)(12,21)(13,18)(14,24)(15,16)(19,20)(22,23)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,38)(39,40);(2,3)(6,7)(10,16)(13,21)(17,24)(20,23)(26,28)(27,34)(32,36)(35,40);(1,3,5,2)(4,7,8,6)(9,11,15,14)(10,17,16,24)(12,22,18,19)(13,20,21,23)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,4,5,8)(2,6,3,7)(9,12,15,18)(10,13,16,21)(11,19,14,22)(17,23,24,20)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,39,40,38);(9,13)(10,18)(11,20)(12,16)(14,23)(15,21)(17,19)(22,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(9,14,15,11)(10,17,16,24)(12,22,18,19)(13,23,21,20)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^2=g,f^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=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022558);