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C₂×C₄²×C₈ #20325

order	256
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 6291456
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂³×C₄³

Aliases

C₂×C₄²×C₈

Subgroups

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

1023 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×C₈×C₄²⋊C₃
maximal subgroup of	C₂×#7015, #94879, #94881, …
maximal normal subgroup of	C₂×#7015, #94879, #94881, …
maximal quotient of	C₂×C₄×C₈², C₂×C₄×#322, C₂×C₈×#97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13313);