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C₈×#127 #21680

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂³
commutator	C₂²	C₂³×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂³	C₂³×C₄
prime radical	C₂×#127	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₈×#127⋊C₃
maximal subgroup of	C₈×#331, #96853, #96856, …
maximal normal subgroup of	C₈×#331, #96853, #96856, …
maximal quotient of	C₈×#492, C₈×#493, C₈×#494, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14668);