#8137

order	256
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₄²
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#376
prime radical	C₂³	#117
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,5,4,6,7,8)(9,10)(11,19,22,32)(12,16)(13,17)(14,18)(15,21,27,35)(20,31,34,39)(23,28)(24,29)(25,30)(26,33,37,40)(36,38);(9,11)(10,15,18,27)(12,20)(13,21)(14,22)(16,26,29,37)(17,19,30,32)(23,33)(24,34)(25,35)(28,31,38,39)(36,40);(9,12,13,23,14,24,25,36)(10,16,17,28,18,29,30,38)(11,20,21,33,22,34,35,40)(15,26,19,31,27,37,32,39)

Polycyclic presentation:

2^8|a^2=e,b^2=f,c^2=d,d^2=g,e^2=h,b^a=bd,f^a=fg,e^b=eg

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	H(3,ℤ/8), #93253, #93254, …
maximal normal subgroup of	H(3,ℤ/8), #93253, #93254, …
maximal quotient of	#96658, #96659, #96660, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1125);