#20516

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂³
commutator	C₂²	C₂²×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴
socle, prime radical	C₂³	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6)(7,11)(8,12)(9,10)(13,14,16,20,17,18,23,25)(15,19,21,26,22,24,27,28);(1,2,4,3)(6,10)(11,12);(1,3,4,2)(5,7,9,8)(6,11,10,12)(14,18)(19,24)(20,25)(26,28);(5,8,9,7)(6,12,10,11)(13,15,16,21,17,22,23,27)(14,19,20,26,18,24,25,28)

Polycyclic presentation:

2^8|a^2=f,b^2=g,c^2=eg,d^2=ef,f^2=h,b^a=be,c^a=ch

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#96775, #96777, #96779, …
maximal normal subgroup of	#96775, #96777, #96779, …
maximal quotient of	#184376, #184378, C₈×#446, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13504);