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C₂×#8886 #192591

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	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1738 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)(7,8,10,13)(9,11,14,12)(15,16,19,22)(17,20,18,21);(3,5,6,4)(8,11)(9,14)(12,13)(16,20)(17,18)(21,22);(16,21)(17,18)(20,22)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#209860, C₄×#8886, D₃×#8886, …
maximal normal subgroup of	C₂×#209860, C₄×#8886, #72279300, …
maximal quotient of	#72224799, #72224804, #72226778, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99787);