Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C22×AGL(1,ℤ/32) (order 2048) · Inn(G) = D16 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | D16 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | C4×D16 |
| prime radical | #2262 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,8,14,29,13)(3,15,18,45,20,46,19,51)(4,10,22,37,23,38,7,30)(5,24,26,54,28,56,60,55)(9,32,34,50,36,49,43,17)(11,21,40,52,42,53,41,27)(16,44,47,63,48,64,31,39)(25,33,57,61,59,62,58,35);(1,3)(2,9)(4,16)(5,25)(6,18)(7,31)(8,20)(10,33)(11,39)(12,34)(13,43)(14,36)(15,24)(17,27)(19,29)(21,32)(22,47)(23,48)(26,57)(28,59)(30,35)(37,61)(38,62)(40,44)(41,64)(42,63)(45,54)(46,56)(49,53)(50,52)(51,55)(58,60);(1,4,6,22,8,23,29,7)(2,10,12,37,14,38,13,30)(3,16,18,47,20,48,19,31)(5,21,26,52,28,53,60,27)(9,33,34,61,36,62,43,35)(11,24,40,54,42,56,41,55)(15,44,45,63,46,64,51,39)(17,25,32,57,50,59,49,58)Polycyclic presentation:
2^8|a^2=e,c^2=e,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #94611, #94612, #151081, … |
| maximal normal subgroup of | #94611, #94612, #151081, … |
| maximal quotient of | C8×D32, C8×SD64, C8×Q64, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6643);