Finite Group Databasebeta

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C₂×#11386 #358777

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	C₂×C₂²⋊C₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(3,4)(5,6)(15,16,18,21,19,17,22,20);(7,8,10,12)(9,13,14,11)(15,17)(16,19)(18,20)(21,22);(4,6)(7,9,10,14)(8,11,12,13)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#358779, #72238886, C₂×#375492
maximal normal subgroup of	C₂×#358779, #72238886, C₂×#375492
maximal quotient of	#72238828

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265973);