C₂×#18788 #6348705

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#3038	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3370 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255901);