Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
535 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,8)(5,7)(10,15,17,32)(12,23)(13,27)(16,31,34,38)(18,24,28,33)(19,35,37,36)(20,39)(21,29)(25,40)(26,30);(1,3,5,4)(2,6,7,8)(9,10)(11,19)(12,24)(13,28)(14,17)(15,30)(16,20)(18,27)(21,31)(22,37)(23,32)(25,35)(26,33)(29,38)(34,40)(36,39);(1,4,5,3)(2,6,7,8);(9,11,14,22)(10,16,17,34)(12,25,26,39)(13,29,27,21)(15,31,32,38)(18,36,28,35)(19,33,37,24)(20,23,40,30)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72313275, #72313289, #72313326, … |
| maximal normal subgroup of | #72313275, #72313289, #72313326, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405178);