D₄×#474 #6574522

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#447
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×D₄²	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2671 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:

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

maximal subgroup of	D₄×#3384, D₄×#3294, D₄×#3115
maximal normal subgroup of	D₄×#3384, D₄×#3294, D₄×#3115

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481718);