C₂×C₄×SD₆₄ #512755

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₄×SD₆₄

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂³	C₂×D₁₆
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

466 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(3,4,5,6);(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38);(8,22)(9,37)(10,20)(11,35)(12,18)(13,33)(14,16)(15,31)(17,29)(19,27)(21,25)(24,38)(26,36)(28,34)(30,32)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₄×#2263, C₂×C₈×SD₆₄, C₂×#151078
maximal normal subgroup of	C₂×C₄×#2263, C₂×C₈×SD₆₄, C₂×#151078
maximal quotient of	C₂×C₈×SD₆₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419951);