C₂×#32704 #7513955

order	512
nilpotency class	3
isoclinism	isoclinic class of #31076
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	#2804
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1514 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);(3,4,6,11,8,12,7,10)(5,13,14,18,16,17,15,9);(3,5)(4,9)(6,14)(7,15)(8,16)(10,17)(11,13)(12,18);(4,10,12,11)(5,14,16,15);(5,14,16,15)(9,17,18,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421151);