D₄×#490 #6579401

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#468
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	D₄×#127	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2722 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)(7,10)(8,9)(11,12)(14,19)(15,20)(16,17);(6,9)(10,12)(13,14,16,19,18,20,17,15);(5,7,8,11)(6,10,9,12)(13,15,16,14,18,19,17,20)

maximal subgroup of	D₄×#3180, D₄×#3366, D₄×#3318
maximal normal subgroup of	D₄×#3180, D₄×#3366, D₄×#3318

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486597);