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C₂×#12718 #493418

order	512
nilpotency class	4
isoclinism	isoclinic class of #12715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	C₂×#3498	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

3748 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)(5,14)(6,16,7,11)(8,12)(9,13)(10,15,18,17);(3,5,9,8)(4,10,13,18)(6,15,7,17)(11,12,16,14);(3,6,9,7)(4,10,13,18)(5,15,8,17)(11,14,16,12)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#511819, #72237532, #72237550, …
maximal normal subgroup of	C₂×#511819, #72237532, #72237550, …
maximal quotient of	#72231096, #72231302, C₂×#147577, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400614);