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Cayley diagram

C4×#452 #21451

order256
nilpotency class3
isoclinismisoclinic class of #447
automorphismAut(G) = C211×D4 (order 16384) · Inn(G) = C2×D4 · Out(G) = C210
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC4×#452

Subgroups

subgroupG/H
centerC22×C4C2×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC22×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,3,4);(5,6)(7,8)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×#452, #208655, #208661, …
maximal normal subgroup ofC8×#452, #208655, #208661, …
maximal quotient ofC8×#452, #208661, #208673, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14439);