C₂×#10601 #242409

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁴	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

918 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(7,8,11,14,10,13,9,12)(16,19)(17,20)(21,22);(3,4)(5,6)(15,16,20,22,18,21,17,19);(4,6)(7,8,10,13)(9,14,11,12)(15,16,18,21)(17,22,20,19)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149605);