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 | |
4336 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)(9,10,14,17)(11,15,19,13)(12,16,20,23)(18,21,24,22)(25,26)(27,29)(28,31)(30,36)(32,34)(33,38)(35,39)(37,40);(2,3)(6,7)(9,11,14,19)(10,13,17,15)(12,18,20,24)(16,22,23,21)(25,27)(26,32)(28,29)(30,35)(31,34)(33,39)(36,37)(38,40);(1,3,5,2)(4,7,8,6)(12,20)(16,23)(18,24)(21,22)(26,28)(27,34)(33,36)(35,40);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,21)(20,24)(22,23)(25,28,31,26)(27,29,34,32)(30,36,38,33)(35,37,40,39);(9,12,14,20)(10,16,17,23)(11,18,19,24)(13,22,15,21)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,40)(38,39);(9,13,14,15)(10,11,17,19)(12,21,20,22)(16,24,23,18)(25,30,31,38)(26,33,28,36)(27,35,34,40)(29,37,32,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=hi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9993546);