Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
441 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,14,17,19)(15,18,22,26)(16,20,23,27)(21,25,24,28);(6,9)(7,12)(15,21)(16,23)(18,25)(20,27)(22,24)(26,28);(2,4)(13,15,17,22)(14,18,19,26)(16,24,23,21)(20,28,27,25);(2,4)(5,7,8,12)(6,10,9,11)(14,19)(18,26)(20,27)(25,28)Polycyclic presentation:
2^8|a^2=eh,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6632841, #6632846, #6632850, … |
| maximal normal subgroup of | #6632841, #6632846, #6632850, … |
| maximal quotient of | #341075, #341076, #341079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23888);