C₂²×#2081 #6432194

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂³	#397
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3306 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);(5,6,9,11,10,14,8,13)(7,15,17,20,16,19,18,12);(6,11)(7,16)(8,9)(12,19)(13,14)(15,20);(5,7,8,17,10,16,9,18)(6,12,13,15,14,20,11,19)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339390);