#17726

order	256
nilpotency class	3
isoclinism	isoclinic class of #17547
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
prime radical	#3511	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=f,c^2=h,d^2=g,e^2=h,f^2=gh,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dg

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#221228, #221229, #221239, …
maximal normal subgroup of	#221228, #221229, #221239, …
maximal quotient of	#160514, #160517, #160518, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10714);