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C₂×C₂²⋊#338 #353298

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

C₂×C₂²⋊#338

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#95
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#359476, #72238750, #72238751, …
maximal normal subgroup of	C₂×#359476, #72238750, #72238751, …
maximal quotient of	#72237153, #72265327, D₃×C₂²⋊#338, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260494);