Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | #392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,15,17,20,25,22)(16,19,24,21,23,26,28,27);(5,7)(6,9)(8,12)(10,11)(13,15,18,20,17,22,25,14)(16,21,24,26,23,27,28,19);(1,2,3,4)(7,12)(9,11)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28);(6,10)(9,11)(14,20)(15,22)(16,23)(24,28)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ceh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22175, #6376131, #6376133, … |
| maximal normal subgroup of | C2×#22175, #6376131, #6376133, … |
| maximal quotient of | #219012, #219013, #219014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15163);