#19229

order	256
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3247	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

303 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,26)(20,25)(21,23)(22,29)(24,27)(28,32)(30,31);(1,3,4,11,6,13,5,12)(2,7,14,16,9,15,8,10)(18,23)(24,31)(25,29)(26,32);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(17,19)(18,24)(20,27)(21,28)(22,30)(23,31)(25,32)(26,29);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,20,21,22)(18,25,23,29)(19,27,28,30)(24,32,31,26)

Polycyclic presentation:

2^8|b^2=egh,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^c=dg,f^c=fh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#19229, #6353718, #6353723, …
maximal normal subgroup of	C₂×#19229, #6353718, #6353723, …
maximal quotient of	#189584, #197772, #197773, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12217);