C₂²×#395 #34534

order	256
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) order 75497472
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3023 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27522);