Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C8×D4 |
| prime radical | C23 | #131 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,11,16,28,25,12,17,29,24,9,14,23,8)(7,18,21,19,20,32,35,34,22,33,27,36,26,13,30,31);(1,2,3,4)(5,7)(6,13)(8,19)(9,26)(10,27)(11,20)(12,22)(14,33)(15,34)(16,18)(17,32)(21,29)(23,35)(24,31)(25,36)(28,30);(5,8,23,14,9,24,29,17,12,25,28,16,11,15,10,6)(7,19,35,33,26,31,21,32,22,36,30,18,20,34,27,13)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=h,e^2=g,g^2=h,b^a=bd,d^b=dh,e^b=eh112 conjugacy classes — view the full character table.
| maximal subgroup of | #94040, #94041, #94042, … |
| maximal normal subgroup of | #94040, #94041, #94042, … |
| maximal quotient of | #129043, #129059, #129072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5652);