Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,14)(12,15)(13,16)(17,18,19,21,20,22,23,24);(2,4)(5,6,7,8)(10,14)(11,12)(15,16);(10,15)(11,12)(14,16)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dh,e^b=eh,d^c=dh,e^c=eh152 conjugacy classes — view the full character table.
| maximal subgroup of | #72321323, #72321376, #72321402, … |
| maximal normal subgroup of | #72321323, #72321376, #72321402, … |
| maximal quotient of | #72322636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12864);