Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10,14,19)(11,21,12,16)(13,20,15,18)(17,22,24,23);(1,2,4,3)(5,6)(7,8)(9,11,14,12)(10,16,19,21)(13,23,15,22)(17,20,24,18);(6,8)(9,12,14,11)(10,17,19,24)(13,23,15,22)(16,20,21,18);(1,3,4,2)(9,13,14,15)(10,18,19,20)(11,22,12,23)(16,24,21,17)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24221, #6442512, #6442521, … |
| maximal normal subgroup of | C2×#24221, #6442512, #6442521, … |
| maximal quotient of | #261208, #261210, #261216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17209);