Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
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 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,21,26,23,28)(19,29,30,25,20,24,31,32);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19,22,30,21,20,23,31)(18,24,27,32,26,29,28,25);(10,14)(13,16)(17,20,21,19)(18,25,26,32)(22,30,23,31)(24,27,29,28);(1,3,4,8)(2,6,5,7)(18,26)(24,29)(25,32)(27,28)Polycyclic presentation:
2^8|a^2=eg,b^2=f,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27752, #6458542, #6458560, … |
| maximal normal subgroup of | C2×#27752, #6458542, #6458560, … |
| maximal quotient of | #360399, #363085, #366416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20740);