C₂×#18621 #6348196

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
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×#3498	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

4546 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);(3,4,7,10)(5,9,6,8)(12,15)(13,17)(16,18);(4,8)(5,6)(9,10)(13,17)(15,18);(11,12)(13,18)(14,16)(15,17);(3,5,7,6)(4,8,10,9)

maximal subgroup of	#72316756, C₂×#6348366
maximal normal subgroup of	#72316756, C₂×#6348366

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255392);