#19874

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	#1721	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

304 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:

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

Polycyclic presentation:

2^8|b^2=gh,c^2=gh,d^2=gh,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×#19874, #6358223, #6358240, …
maximal normal subgroup of	C₂×#19874, #6358223, #6358240, …
maximal quotient of	#220767, #220871, #222495, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12862);