D₄×SD₆₄ #513166

order	512
nilpotency class	5
isoclinism	isoclinic class of D₄×D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₄×SD₆₄

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	D₄×D₁₆	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2263, S₄×SD₆₄
maximal normal subgroup of	D₄×#2263

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420362);