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 | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
485 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)(3,5)(4,6)(7,8)(9,10,15,21,16,22,14,30,17,23,33,13,12,19,31,20)(11,24,28,18,25,38,27,37,29,32,34,26,36,40,35,39);(1,3,4,7)(2,5,6,8)(9,11)(10,18)(12,25)(13,32)(14,34)(15,35)(16,36)(17,29)(19,37)(20,38)(21,24)(22,39)(23,26)(27,33)(28,31)(30,40);(2,6)(5,8)(9,12,17,16)(10,19,23,22)(11,25,29,36)(13,30,21,20)(14,15,31,33)(18,37,26,39)(24,38,32,40)(27,34,35,28);(9,13,33,19,16,20,31,10,17,21,15,22,12,30,14,23)(11,26,34,40,25,39,35,24,29,18,28,38,36,37,27,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=eh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,d^b=de,e^b=eh,g^b=ghi,h^b=hi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72313642, #72313651, #72313701, … |
| maximal normal subgroup of | #72313642, #72313651, #72313701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404981);