#19723

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
prime radical	#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

339 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:

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

Polycyclic presentation:

2^8|a^2=g,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×#19723, #6357363, #6357371, …
maximal normal subgroup of	C₂×#19723, #6357363, #6357371, …
maximal quotient of	#191522, #206261, #206266, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12711);