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C₈² #321

order	64
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = GL(2,ℤ/8) (order 1536) · Inn(G) = C₁ · Out(G) = GL(2,ℤ/8)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₈

Aliases

C₈²

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂²
socle, prime radical	C₂²	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

37 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

center of	#7052, C₈×M₃₂, #11410, …
derived subgroup of	C₈²⋊C₃, C₂×C₈²⋊C₃, #1454⋊C₃, …
Frattini subgroup of	C₁₆², #7052, #7053, …
Fitting subgroup of	C₈²⋊C₃, #67992
maximal subgroup of	C₈×C₁₆, #1311, #1312, …
maximal normal subgroup of	C₈×C₁₆, #1311, #1312, …
maximal quotient of	#1273, C₈×C₁₆, #1311, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 2);