D₄² #545

order	64
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) = #72215688 (order 2048) · Inn(G) = C₂⁴ · Out(G) = #2196
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

D₄²

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

214 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 4 generators on 8 points:

(6,8);(5,6)(7,8);(2,4);(1,2)(3,4)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 226);