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C₂×#21236 #6372000

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#6442768, C₂×#6443244, C₂×C₂²⋊C₄×AGL(1,ℤ/8), …
maximal normal subgroup of	C₂×#6442768, C₂×#6443244, C₂×C₂²⋊C₄×AGL(1,ℤ/8), …
maximal quotient of	C₂×#357515

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279196);