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Cayley diagram

C4×M64 #13548

order256
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) order 4096
homologyAbelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C2×C4

Aliases

direct productC4×M64

Subgroups

subgroupG/H
centerC4×C16C22
commutatorC2C2×C4×C16
Frattini, cosocleC2×C16C23
socleC22C22×C16
prime radicalC23C2×C16
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C4×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36);(6,22)(8,24)(10,26)(12,28)(14,30)(16,32)(18,34)(20,36)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#93991, #93992, #93993, …
maximal normal subgroup of#93991, #93992, #93993, …
maximal quotient ofC8×M64, #149499, #149507, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6536);