C₂²×#2392 #7627693

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

	subgroup	G/H
center	C₂⁴×C₄	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁶
socle	C₂⁵	C₂⁴
prime radical	C₂²×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

16290 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534889);