Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187940 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #62819 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,8)(9,10)(11,17,19,15)(12,14)(13,20)(16,21)(18,22,24,23);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24);(1,3)(2,5)(4,7)(6,8)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72303967, #72304001, #72304003, … |
| maximal normal subgroup of | #72303967, #72304001, #72304003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95138);