Finite Group Databasebeta

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Cayley diagram

C4×#479 #21612

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×#479

Subgroups

subgroupG/H
centerC22×C4C2×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socle, prime radicalC23C22×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×#479, #212311, #212317, …
maximal normal subgroup ofC8×#479, #212311, #212317, …
maximal quotient ofC8×#479, #212317, #212345, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14600);