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Cayley diagram

C4×#485 #21577

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) = C24

Aliases

direct productC4×#485

Subgroups

subgroupG/H
centerC22×C4C2×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC24C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2,3,4);(5,6)(7,10)(8,9)(11,12)(13,14,17,21)(15,18,23,27)(16,19,24,20)(22,25,28,26);(6,9)(10,12)(13,15,17,23)(14,18,21,27)(16,22,24,28)(19,25,20,26);(5,7,8,11)(6,10,9,12)(14,19)(16,24)(18,25)(20,21)(22,28)(26,27)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×#485, #212023, #212029, …
maximal normal subgroup ofC8×#485, #212023, #212029, …
maximal quotient ofC8×#485, #212029, #212057, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14565);