#19861

order	256
nilpotency class	3
isoclinism	isoclinic class of #19755
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	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3303	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

298 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#19861, #6358214, #6358235, …
maximal normal subgroup of	C₂×#19861, #6358214, #6358235, …
maximal quotient of	#220757, #220866, #222468, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12849);