Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,20)(17,26)(18,23)(19,24)(21,25)(27,28);(1,2)(3,4)(6,9)(10,12)(13,15,16,23)(14,18,20,22)(17,24,25,27)(19,26,28,21);(2,4)(7,11)(10,12)(13,15)(14,19)(16,23)(17,27)(18,21)(20,28)(22,26)(24,25);(5,7,8,11)(6,10,9,12)(15,23)(18,22)(19,28)(24,27)Polycyclic presentation:
2^8|b^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^b=dh,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29677, #6435010, #6435017, … |
| maximal normal subgroup of | C2×#29677, #6435010, #6435017, … |
| maximal quotient of | #328383, #328386, #328389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22665);