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C₂×#12595 #492924

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

790 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);(7,8,11,16,10,13,19,22,12,17,21,14,20,15,9,18);(3,4,5,6)(8,13,17,15)(14,18,16,22);(7,9,10,11,12,19,20,21)(8,14,15,22,17,16,13,18)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213041, C₂×#124917, #72216658, …
maximal normal subgroup of	#72213041, C₂×#124917, #72216658, …
maximal quotient of	C₂×#146308

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400120);