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 | |
4826 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)(25,26,30,32)(27,34,37,31)(28,33,38,40)(29,36,39,35);(2,5)(6,7)(10,14)(15,17)(16,21)(22,23)(25,27)(26,31)(28,36)(29,33)(30,37)(32,34)(35,38)(39,40);(1,2)(3,6)(4,5)(7,8)(12,20)(16,21)(18,24)(22,23)(26,32)(29,39)(31,34)(33,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,38)(26,33,32,40)(27,36,37,35)(29,34,39,31);(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,34)(28,35)(29,33)(30,37)(31,32)(36,38)(39,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,35)(27,33)(28,34)(30,39)(31,38)(32,36)(37,40)Polycyclic presentation:
2^9|a^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 | #72341056, #72341139, #72341143, … |
| maximal normal subgroup of | #72341056, #72341139, #72341143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8832400);