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×#509 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,8)(5,7)(9,10,13,17,14,19,24,18)(11,15,20,12,21,22,16,23);(1,3,5,4)(2,6,7,8)(10,11)(12,15)(13,24)(16,17)(18,20)(19,21)(22,23);(1,4,5,3)(2,6,7,8);(9,11)(10,15)(12,17)(13,20)(14,21)(16,24)(18,23)(19,22)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=h,g^2=i,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 | #72313376, #72313420, #72313439, … |
| maximal normal subgroup of | #72313376, #72313420, #72313439, … |
| maximal quotient of | #72343510, #72343543, #72343575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405181);