SD₃₂ #112

order	32
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) = #442 (order 64) · Inn(G) = D₈ · Out(G) = C₄
holomorph	G ⋊ Aut(G) = #72214534
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

SD₃₂

	subgroup	G/H
center, socle	C₂	D₈
commutator, Frattini, cosocle	C₈	C₂²
prime radical	D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

13 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

2^5|a^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,c^b=cd,d^b=de

Here ζ is a primitive root of unity of order 16 (ζ¹⁶ = 1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32, 19);