#7557

order	256
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) order 262144
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂³
commutator	C₂³	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

1443 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 20 points:

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

Polycyclic presentation:

2^8|a^2=g,b^2=h,c^2=e,b^a=bd,c^a=ce,c^b=cf

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 545);