Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
561 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)(13,14,18,23)(15,24,16,20)(17,22,19,27)(21,25,28,26);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,18)(17,26)(19,25)(21,27)(22,28)(23,24);(2,4)(13,16,18,15)(14,21,23,28)(17,26,19,25)(20,22,24,27);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,23,27)(15,25,16,26)(20,21,24,28)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^b=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23585, #6432026, #6432033, … |
| maximal normal subgroup of | C2×#23585, #6432026, #6432033, … |
| maximal quotient of | #241723, #241726, #241727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16573);