C₂²×#2091 #6432397

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center	C₂²×C₄	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	#399
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3630 conjugacy classes of subgroups; only the 914 normal subgroups are shown.

normal subgroups only

This group has 3630 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);(5,6,8,12,9,11,14,18)(7,13,15,19,16,20,17,10);(5,7,14,15,9,16,8,17)(6,10,18,13,11,19,12,20);(6,11)(12,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339593);