C₂²×#1597 #6346337

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#447	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

4838 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);(6,10)(7,8)(11,12)(17,20);(5,6)(7,10)(8,12)(9,11)(13,14)(15,17)(16,18)(19,20);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253533);