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 | |
4599 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,21)(14,15)(16,23)(17,20)(18,24)(19,22)(25,26,30,31)(27,32,35,38)(28,36,37,29)(33,39,40,34);(1,2,5,4)(3,7,8,6)(9,11,14,17)(10,12,15,20)(13,22,23,18)(16,24,21,19)(26,31)(29,36)(32,38)(34,39);(2,4)(6,7)(10,15)(11,17)(16,21)(18,22)(25,27,30,35)(26,32,31,38)(28,33,37,40)(29,34,36,39);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,22)(23,24)(25,28)(26,29)(27,33)(30,37)(31,36)(32,34)(35,40)(38,39);(1,3,5,8)(2,6,4,7)(9,13,14,23)(10,16,15,21)(11,18,17,22)(12,19,20,24)(25,29)(26,28)(27,34)(30,36)(31,37)(32,33)(35,39)(38,40);(1,4,5,2)(3,7,8,6)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,27)(26,32)(28,33)(29,34)(30,35)(31,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8161924);