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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
367 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,32)(29,31);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,27,29,21,20,22,28)(18,23,30,32,25,24,26,31);(10,14)(13,15)(17,20,21,19)(18,24,25,23)(22,28,27,29)(26,31,30,32);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(18,25)(23,24)(26,30)(31,32)Polycyclic presentation:
2^8|b^2=fg,c^2=h,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29041, #6460384, #6460413, … |
| maximal normal subgroup of | C2×#29041, #6460384, #6460413, … |
| maximal quotient of | #362579, #363661, #367811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22029);