#19547

order	256
nilpotency class	3
isoclinism	isoclinic class of #19504
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	#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

276 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#19547, #6354639, #6354643, …
maximal normal subgroup of	C₂×#19547, #6354639, #6354643, …
maximal quotient of	#189619, #197923, #198010, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12535);