Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17,14,16,12,22)(11,18,21,23,19,24,20,15);(1,3)(2,5,6,7)(4,8)(9,11)(10,15,16,23)(12,20)(13,21)(14,19)(17,18,22,24);(11,19)(15,23)(18,24)(20,21);(10,16)(15,23)(17,22)(18,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=h,f^2=h,b^a=be,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235941, #72235964, #72236000, … |
| maximal normal subgroup of | #72235941, #72235964, #72236000, … |
| maximal quotient of | #72264585, C2×#181776 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88972);