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 | |
4452 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)(9,10,13,14)(11,15,19,17)(12,20,21,16)(18,23,24,22)(25,26)(27,34)(28,36)(29,31)(30,33)(32,37)(35,39)(38,40);(2,3)(6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,29)(26,31,33,34)(28,35,37,38)(32,39,36,40);(1,3,5,2)(4,7,8,6)(10,14)(15,17)(16,20)(22,23);(1,4,5,8)(2,6,3,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28)(26,32)(27,35)(29,38)(30,37)(31,39)(33,36)(34,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,29,30,27)(26,31,33,34)(28,35,37,38)(32,40,36,39);(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=hi,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10090315);