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C₂×#8924 #193181

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1818 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)(8,11)(9,13)(12,14)(15,16,18,21)(17,19,22,20);(3,5,6,4)(7,8)(9,14)(10,12)(11,13);(9,13)(11,14)(16,19)(17,22)(20,21)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#209860, #72279385, C₂×#229439, …
maximal normal subgroup of	C₂×#209860, #72279385, C₂×#229439, …
maximal quotient of	#72224808, #72226755, #72226784, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100377);