Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#72283979 (order 4096) · Inn(G) = C22 · Out(G) = C2×#233610 |
| homology | Abelianization (H1) = C8×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C4×C16 | C22 |
| commutator | C2 | C8×C16 |
| socle, prime radical | C22 | C4×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(9,10,11,16,12,17,19,28,13,18,20,29,22,30,31,38,14,15,21,25,23,26,32,35,24,27,33,36,34,37,39,40);(1,2,3,5,4,6,7,8)(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40)Polycyclic presentation:
2^8|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,g^2=h,b^a=bh160 conjugacy classes — view the full character table.
| Frattini subgroup of | #72230688, #72230689, #72230702, … |
| maximal subgroup of | #93666, #93746, #93749, … |
| maximal normal subgroup of | #93666, #93746, #93749, … |
| maximal quotient of | #93664, #93665, #93989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 317);