C₂×#10439 #238669

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1250 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);(4,5)(7,8,9,11)(10,12,14,13)(16,19)(20,21);(3,4)(5,6)(8,11)(12,13)(15,16,17,19)(18,20,22,21);(3,5,6,4)(8,12)(10,14)(11,13)(16,20)(18,22)(19,21)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145865);