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 | |
4838 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26,31,33)(27,34,28,32)(29,30,39,40)(35,38,37,36);(2,5)(6,7)(10,14)(15,17)(16,21)(22,23)(25,27,31,28)(26,32,33,34)(29,35,39,37)(30,36,40,38);(1,2)(3,6)(4,5)(7,8)(12,20)(16,21)(18,24)(22,23)(26,33)(30,40)(32,34)(36,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,31,27)(26,32,33,34)(29,35,39,37)(30,38,40,36);(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,29)(26,30)(27,35)(28,37)(31,39)(32,36)(33,40)(34,38);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,29)(27,36)(28,38)(31,40)(32,35)(33,39)(34,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72340990, #72341086, #72341090, … |
| maximal normal subgroup of | #72340990, #72341086, #72341090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8832504);