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Cayley diagram

C4×C64 #7509

order256
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×#496097 (order 1024) · Inn(G) = C1 · Out(G) = C2×#496097
homologyAbelianization (H1) = G · Schur multiplier (H2) = C4

Aliases

abelianC4×C64

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2×C32C22
socle, prime radicalC22C2×C32
LCSG ▷ C1
UCSC1 ◁ G
LECSG ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

43 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 68
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 2 generators on 68 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64);(65,66,67,68)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of#72277918, #72278050
maximal subgroup ofC8×C64, #94504, #94507, …
maximal normal subgroup ofC8×C64, #94504, #94507, …
maximal quotient ofC8×C64, #94507, #94755, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 497);