Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = #72287717 (order 8192) · Inn(G) = D16 · Out(G) = C8×#120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D16 |
| commutator | C16 | C42 |
| Frattini, cosocle | C22×C16 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
95 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,4)(5,6,7,8)(10,16)(11,12)(13,21)(14,27)(17,29)(18,33)(19,32)(20,31)(22,38)(23,26)(24,25)(28,37)(30,35)(34,39)(36,40);(2,4)(9,10,12,17,13,18,24,32,27,35,38,40,37,39,23,31,15,20,26,34,28,36,22,30,14,19,25,33,21,29,11,16)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,e^2=f,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fg,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #94690, #94691, C2×#7471, … |
| maximal normal subgroup of | #94690, #94691, C2×#7471, … |
| maximal quotient of | #94021, #94044, #94202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 459);