Finite Group Databasebeta

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C₂×#12725 #493455

order	512
nilpotency class	4
isoclinism	isoclinic class of #12725
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#3490	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2812 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#493474, #72219393, #72225782, …
maximal normal subgroup of	C₂×#493474, #72219393, #72225782, …
maximal quotient of	#72219304, #72219305, #72219325, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400651);