C₂²×SD₃₂ #3409

order	128
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×SD₃₂

	subgroup	G/H
center, socle	C₂³	D₈
commutator, Frattini, cosocle	C₈	C₂⁴
prime radical	C₂²×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

200 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2);(3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20);(6,12)(7,19)(8,10)(9,17)(11,15)(14,20)(16,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2141);