Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| 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 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
693 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)(3,4)(9,10,14,17,13,16,21,24)(11,12,19,20,18,15,23,22);(2,4)(5,6)(7,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,24)(21,23);(6,8)(9,12,13,15)(10,11,16,18)(14,22,21,20)(17,23,24,19);(2,4)(10,16)(11,18)(17,24)(19,23)Polycyclic presentation:
2^8|a^2=f,c^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24392, #6437747, #6437755, … |
| maximal normal subgroup of | C2×#24392, #6437747, #6437755, … |
| maximal quotient of | #268767, #268769, #268775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17380);