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 | |
4630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,18)(12,20)(13,17)(14,15)(16,21)(19,23)(22,24)(25,26,31,28)(27,32,34,30)(29,33,39,36)(35,40,37,38);(3,7)(6,8)(10,15)(11,17)(12,21)(23,24)(25,27,31,34)(26,32,28,30)(29,37,39,35)(33,38,36,40);(1,2)(3,6)(4,5)(7,8)(10,15)(13,18)(16,20)(23,24)(27,34)(30,32)(35,37)(38,40);(1,3,4,7)(2,6,5,8)(9,11,14,17)(10,13,15,18)(12,22,21,19)(16,24,20,23)(25,28,31,26)(27,30,34,32)(29,33,39,36)(35,40,37,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,21)(10,16,15,20)(11,19,17,22)(13,23,18,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37);(9,13)(10,17)(11,15)(12,23)(14,18)(16,22)(19,20)(21,24)(25,30)(26,27)(28,34)(29,38)(31,32)(33,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8266937);