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C₂×#7983 #181237

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231716, #72231722, C₂×#234325, …
maximal normal subgroup of	#72231716, #72231722, C₂×#234325, …
maximal quotient of	#72260989, #72264337, #72264585, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88433);