#19569

order	256
nilpotency class	3
isoclinism	isoclinic class of #19552
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	#3207	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

301 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#19569, #6354670, #6354699, …
maximal normal subgroup of	C₂×#19569, #6354670, #6354699, …
maximal quotient of	#189167, #197905, #197991, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12557);