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C₂×#8153 #181963

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

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

2010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#208872, C₂×#208920, C₂×#181965, …
maximal normal subgroup of	C₂×#208872, C₂×#208920, C₂×#181965, …
maximal quotient of	C₂×#128382, C₂×#128384

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89159);