D₄×#455 #6573665

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#453
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	D₄×H(5,𝔽₂)	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4893 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480861);