C₂×#10193 #237371

order	512
nilpotency class	3
isoclinism	isoclinic class of #10192
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3205	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1644 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);(4,9)(5,10)(6,7)(11,12)(13,16)(14,18)(15,17);(3,4)(5,8)(6,10)(7,9)(12,16)(13,15)(17,18);(3,5,8,4)(6,10,7,9)(11,12,14,18)(13,16,15,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144567);