C₂²×#2031 #6431841

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center	C₂²×C₄	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	#393
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4162 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)(7,14,16,20)(8,12)(9,18)(10,13)(11,17,19,15);(7,15,16,17)(11,14,19,20);(5,7)(6,11)(8,15)(9,17)(10,16)(12,20)(13,19)(14,18)

maximal subgroup of	#72262963, #72262969, #72262977, …
maximal normal subgroup of	#72262963, #72262969, #72262977, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339037);