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C₄×C₈×Q₈ #20515

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

C₄×C₈×Q₈

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴
socle, prime radical	C₂³	C₂³×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2,3,4);(5,6,7,8,9,10,11,12);(13,14,15,16)(17,20,19,18);(13,17,15,19)(14,18,16,20)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₄×C₈×SL(2,𝔽₃)
maximal subgroup of	C₈×Q₈⋊C₈, #96687, #96689, …
maximal normal subgroup of	C₈×Q₈⋊C₈, #96687, #96689, …
maximal quotient of	C₈²×Q₈, #184372, #184373, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13503);