Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
373 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)(4,6)(5,8)(17,18,19,23,20,24,21,22);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,20,21)(18,22,24,23);(10,13)(14,16)(18,23)(19,21)(22,24);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,24)(22,23)Polycyclic presentation:
2^8|a^2=fh,b^2=e,d^2=gh,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29433, #6461597, #6461625, … |
| maximal normal subgroup of | C2×#29433, #6461597, #6461625, … |
| maximal quotient of | #363305, #364297, #371281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22421);