C₂²×#2069 #6432149

order	512
nilpotency class	3
isoclinism	isoclinic class of #2069
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂³	#396
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

3570 conjugacy classes of subgroups; only the 914 normal subgroups are shown.

normal subgroups only

This group has 3570 conjugacy classes of subgroups — too many to draw. Only its 914 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339345);