C₂×#29384 #6461444

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	#2598
prime radical	C₂×#2010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2046 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 18 points:

(1,2);(3,4)(5,12)(6,9)(7,11)(8,13)(10,15)(14,17)(16,18);(3,5,6,13,7,14,15,18)(4,8,10,12,11,16,9,17);(8,16)(12,17);(4,9,11,10)(8,17,16,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368640);