Finite Group Databasebeta

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C₂×#12582 #492897

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3346 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);(7,8)(9,15,16,21)(10,18,19,13)(11,14)(12,17)(20,22);(3,4,5,6)(7,9)(8,12,14,20)(10,17)(11,16)(13,15,18,21)(19,22);(9,16)(12,20)(15,21)(17,22)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#498886, #72233189, #72233200, …
maximal normal subgroup of	C₂×#498886, #72233189, #72233200, …
maximal quotient of	#72233041, #72237444, D₃×#12582, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400093);