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 | #37290 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,13)(12,20)(14,15)(16,21)(17,18)(19,24)(22,23)(25,26,30,34)(27,33,29,31)(28,32,38,36)(35,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,22)(20,24)(21,23)(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(2,5)(6,8)(10,15)(11,17)(12,21)(22,24)(25,26,30,34)(27,31,29,33)(28,36,38,32)(35,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,17)(14,15)(19,24)(20,21)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,21)(10,16,15,20)(11,19,17,23)(13,22,18,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,3,4,7)(2,6,5,8)(9,13,14,18)(10,17,15,11)(12,22,21,24)(16,23,20,19)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8039712);