Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62816 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,14,17,15,11,13,18)(12,16,22,23,20,19,21,24);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,20)(21,23)(22,24);(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,23)(18,24);(12,20)(16,19)(21,22)(23,24);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,17,11,18)(12,21,20,22)(16,23,19,24)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72287778, #72320015, #72320117 |
| maximal normal subgroup of | #72287778, #72320015, #72320117 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250389);