Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C2×D4 · Out(G) = C25×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,17,14,11,18,16)(12,15,20,23,21,19,24,22);(1,3,5,4)(2,6,7,8);(9,11)(10,14)(12,19)(13,16)(15,21)(17,18)(20,22)(23,24);(1,4,5,3)(2,6,7,8)(9,12,13,20,14,21,18,24)(10,15,17,23,11,19,16,22)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=eg,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,f^c=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #136575, #136583, #137596, … |
| maximal normal subgroup of | #136575, #136583, #137596, … |
| maximal quotient of | C8×#474, C8×#476, C8×#475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14615);