#19722

order	256
nilpotency class	3
isoclinism	isoclinic class of #19713
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

373 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)(19,28)(20,24)(21,26)(22,31)(23,27)(25,29)(30,32);(1,3,4,11,6,13,5,12)(2,7,14,16,9,15,8,10)(17,19,21,30,23,29,22,20)(18,24,31,25,27,32,26,28);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(17,20,22,29,23,30,21,19)(18,25,26,24,27,28,31,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,21,23,22)(18,26,27,31)(19,20,29,30)(24,25,32,28)

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#19722, #6357363, #6357368, …
maximal normal subgroup of	C₂×#19722, #6357363, #6357368, …
maximal quotient of	#191522, #206262, #206264, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12710);