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C₂×#8977 #193971

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#459	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2230 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);(4,6)(7,8,9,12,10,13,14,11)(16,19,20,22);(3,4,5,6)(8,11)(9,14)(12,13)(17,21)(19,22);(15,16)(17,22)(18,20)(19,21)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#209860, D₃×#8977, #72279328, …
maximal normal subgroup of	C₂×#209860, #72279328, C₂×#210060, …
maximal quotient of	#72224798, #72224805, #72226780, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101167);