#12593

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) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₄²
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×#97
prime radical	C₂³	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

169 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)(3,6)(4,5)(7,8)(9,10,14,20,15,22,30,37,16,21,29,36,13,19,12,18)(11,23,26,39,27,17,24,33,28,35,31,32,25,34,40,38);(2,5)(6,8)(9,11)(10,17,21,34)(12,24)(13,25)(14,31)(15,27)(16,28)(18,32,37,39)(19,23,22,35)(20,38,36,33)(26,29)(30,40);(1,3,4,7)(2,6,5,8)(9,12,13,29,16,30,15,14)(10,18,19,36,21,37,22,20)(11,24,25,26,28,40,27,31)(17,32,23,33,34,39,35,38)

Polycyclic presentation:

2^8|a^2=e,b^2=f,c^2=d,d^2=h,e^2=g,g^2=h,b^a=bd,f^a=fh,e^b=eh

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	#128206, #128207, #128211, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5581);