#18707

order	256
nilpotency class	3
isoclinism	isoclinic class of #18682
automorphism	Aut(G) order 4096
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	#3547	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

437 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)(3,11,14,15)(4,9,5,8)(7,12,16,13)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32);(1,3,6,14)(2,7,10,16)(4,12,5,13)(8,15,9,11)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(1,4,6,5)(2,9,10,8)(3,12,14,13)(7,11,16,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#18707, #6348312, #6348316, …
maximal normal subgroup of	C₂×#18707, #6348312, #6348316, …
maximal quotient of	#167786, #167816, #167856, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11695);