C₂²×#2048 #6431930

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5134 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);(6,11)(7,14)(8,9)(12,19)(13,17)(15,16);(5,6)(7,12)(8,17)(9,11)(10,13)(14,18)(15,20)(16,19);(5,7,8,15,10,16,9,14)(6,12,11,18,13,19,17,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339126);