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C₄²×C₈ #1724

order	128
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 98304
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₄³

Aliases

C₄²×C₈

Subgroups

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

204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

center of	C₈×#322, #94852, C₄×#1311, …
Frattini subgroup of	#92817, #92818, #92862, …
Fitting subgroup of	C₈×C₄²⋊C₃, #11710952, C₈×#857, …
maximal subgroup of	#7015, #7148, C₄×C₈², …
maximal normal subgroup of	#7015, #7148, C₄×C₈², …
maximal quotient of	C₄×C₈², C₄×#322, C₈×#97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 456);