#19071

order	256
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	Aut(G) = C₂⁶×#120 (order 2048) · Inn(G) = C₂×#120 · Out(G) = C₂⁵
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3553	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

405 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,28)(19,29,32,41)(20,33,34,42)(21,27,22,26)(24,36,39,31)(25,37,40,35)(30,38,44,48)(43,45,47,46);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,41)(27,29)(28,39)(33,47)(34,44)(35,46)(37,45)(40,48)(42,43);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20,23,34)(18,25,28,40)(19,30,32,44)(21,35,22,37)(24,38,39,48)(26,33,27,42)(29,43,41,47)(31,45,36,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(20,34)(25,40)(30,44)(33,42)(35,37)(38,48)(43,47)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#19071, #6351479, #6351481, …
maximal normal subgroup of	C₂×#19071, #6351479, #6351481, …
maximal quotient of	#167788, #167790, #167817, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12059);