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 | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
687 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)(3,5)(4,6)(7,8)(9,10,13,17,14,19,24,18)(11,15,20,12,21,22,16,23);(1,3,4,7)(2,5,6,8)(10,11)(12,15)(13,24)(16,17)(18,20)(19,21)(22,23);(2,6)(5,8);(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,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 | #72342431, #72342568, #72343223, … |
| maximal normal subgroup of | #72342431, #72342568, #72343223, … |
| maximal quotient of | #72343352, #72343418, #72343621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404901);