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Cayley diagram

C23×C8 #565

order64
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×#72209833 (order 43008) · Inn(G) = C1 · Out(G) = C2×#72209833
homologyAbelianization (H1) = G · Schur multiplier (H2) = C26

Aliases

abelianC23×C8

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC4C24
socle, prime radicalC24C4
LCSG ▷ C1
UCSC1 ◁ G
LECSG ▷ C4 ▷ C2 ▷ C1

169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 14 points:

(1,2,3,4,5,6,7,8);(9,10);(11,12);(13,14)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

center ofC8×#95, C2×#1374, C2×#1726, …
derived subgroup of#142232, #142233, #142234, …
Frattini subgroup of#7183, #7184, #7185, …
Fitting subgroup ofC2×C8×A4, C2×A4⋊C16, #73104, …
maximal subgroup ofC2×#336, #1729, C2×#340, …
maximal normal subgroup ofC2×#336, #1729, C2×#340, …
maximal quotient ofC22×C4×C8, C22×C22⋊C8, C22×C4⋊C8, …
outer automorphism group ofD4⋊C32, #7383, #7384, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 246);