C₂²⋊C₄×#124 #519381

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6354 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 426577);