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C₈×C₃₂ #7328

order	256
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×#72283979 (order 4096) · Inn(G) = C₁ · Out(G) = C₂×#72283979
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₈

Aliases

C₈×C₃₂

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₁₆	C₂²
socle, prime radical	C₂²	C₄×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

67 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 40 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)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

center of	#72234245, #72234298, C₈×M₁₂₈, …
Frattini subgroup of	#72234393, C₁₆×M₁₂₈, #72234395, …
maximal subgroup of	C₁₆×C₃₂, #93664, #93665, …
maximal normal subgroup of	C₁₆×C₃₂, #93664, #93665, …
maximal quotient of	C₁₆×C₃₂, #93665, #93989, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 316);