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 | |
4731 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)(3,5)(4,7)(6,8)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26,30,34)(27,31,35,40)(28,32,38,37)(29,33,39,36);(2,3)(6,7)(10,15)(12,16)(17,22)(20,23)(27,35)(28,38)(31,40)(32,37);(1,2,5,3)(4,6,8,7)(13,21)(17,22)(18,24)(20,23)(25,27)(26,31)(28,36)(29,32)(30,35)(33,38)(34,40)(37,39);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24);(1,4,5,8)(2,6,3,7)(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29)(26,33)(27,37)(28,31)(30,39)(32,35)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=h,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=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109569);