Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
394 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)(7,11)(8,9)(10,12)(13,14,17,22,18,23,16,21)(15,19,24,20,26,28,25,27);(1,2)(3,4)(6,9)(10,11)(13,15,16,25,18,26,17,24)(14,19,21,27,23,28,22,20);(2,4)(13,15,18,26)(14,20,23,27)(16,24,17,25)(19,21,28,22);(5,7,8,12)(6,10,9,11)(13,16,18,17)(14,21,23,22)(15,24,26,25)(19,20,28,27)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25311, #6448107, #6448113, … |
| maximal normal subgroup of | C2×#25311, #6448107, #6448113, … |
| maximal quotient of | #286976, #286978, #286984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18299);