Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480690 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35513 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16,20,14)(13,17)(18,22)(19,23,24,21);(1,3)(2,5)(4,8)(6,7)(11,18)(15,22)(19,24)(21,23);(3,8)(5,7)(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(2,6)(5,7)(9,11,13,18)(10,15,17,22)(12,19,20,24)(14,21,16,23)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^c=fi,g^c=ghi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72226017, #72226081, #72238365, … |
| maximal normal subgroup of | #72226017, #72226081, #72238365, … |
| maximal quotient of | #72226008, #72237600, #72238369 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403396);