Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1602 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)(3,7,8,5)(4,6)(9,10)(11,13)(12,14)(15,16)(17,18,20,22,21,24,19,23);(1,3)(2,5)(4,8)(6,7)(10,13)(11,16)(14,15);(2,6)(5,7)(10,14)(13,15)(17,19,21,20)(18,22,24,23);(10,14)(13,15)(17,19,21,20)(18,23,24,22)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cgi,d^a=dh,e^a=eg,e^b=eh,f^b=fgh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309144, #72309152, #72309180, … |
| maximal normal subgroup of | #72309144, #72309152, #72309180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 68440);