Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
386 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)(7,12)(8,10)(9,11)(13,14)(15,19)(16,18)(17,20)(21,28)(22,25)(23,26)(24,27);(5,7,9,8)(6,10,11,12)(14,18)(15,21)(17,23)(19,27)(22,24)(25,28);(2,4)(5,8,9,7)(6,10,11,12)(13,15,16,22)(14,19,18,25)(17,24,23,21)(20,27,26,28);(6,11)(10,12)(14,18)(15,22)(20,26)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=eh,c^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24590, #6453071, #6453077, … |
| maximal normal subgroup of | C2×#24590, #6453071, #6453077, … |
| maximal quotient of | #275323, #275324, #275325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17578);