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 | #37839 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,14,18)(11,15,19,22)(12,16,21,24)(13,17,23,20)(25,26)(27,36)(28,32)(29,34)(30,35)(31,38)(33,37)(39,40);(3,4)(6,8)(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(25,27,30,38)(26,31,35,36)(28,29,37,39)(32,40,33,34);(1,3,5,4)(2,6,7,8)(11,19)(12,21)(15,22)(16,24)(25,28,30,37)(26,32,35,33)(27,29,38,39)(31,40,36,34);(3,4)(6,8)(9,12)(10,16)(11,20)(13,15)(14,21)(17,19)(18,24)(22,23)(25,28)(26,33)(27,29)(30,37)(31,34)(32,35)(36,40)(38,39);(1,4,5,3)(2,6,7,8)(9,13)(10,17)(11,16)(12,22)(14,23)(15,21)(18,20)(19,24)(25,29)(26,34)(27,37)(28,38)(30,39)(31,32)(33,36)(35,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(25,26)(27,31)(28,33)(29,34)(30,35)(32,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812746);