#19585

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	#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

275 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)(3,11,13,14)(4,8,7,16)(5,12,15,9)(17,18,22,26)(19,23,28,32)(20,25,21,30)(24,29,31,27);(1,3,5,8,6,13,15,16)(2,7,9,11,10,4,12,14)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(1,4)(2,8)(3,12)(5,14)(6,7)(9,13)(10,16)(11,15)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(3,13)(4,7)(8,16)(11,14)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,24,32,31)

Polycyclic presentation:

2^8|a^2=gh,b^2=e,d^2=h,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×#19585, #6354695, #6354697, …
maximal normal subgroup of	C₂×#19585, #6354695, #6354697, …
maximal quotient of	#189168, #197910, #197992, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12573);