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Cayley diagram

C2×M32 #503

order64
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×#3025 (order 256) · Inn(G) = C22 · Out(G) = C23×D4
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C22

Aliases

direct productC2×M32

Subgroups

subgroupG/H
centerC2×C8C22
commutatorC2C22×C8
Frattini, cosocleC8C23
socleC22C2×C8
prime radicalC23C8
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ G
LECSG ▷ C8 ▷ C4 ▷ C2 ▷ C1

45 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2);(3,4,5,9,6,10,11,16,7,8,12,14,13,15,17,18);(4,8)(9,14)(10,15)(16,18)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of#93768, #93769, #93780, …
Frattini subgroup of#7336, #7337, #7364, …
maximal subgroup of#1375, #1376, #1377, …
maximal normal subgroup of#1375, #1376, #1377, …
maximal quotient ofC2×#346, C4×M32, C2×C22⋊C16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 184);