C₂×#10518 #238863

order	512
nilpotency class	3
isoclinism	isoclinic class of #10518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3245	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1256 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146059);