#20265

order	256
nilpotency class	3
isoclinism	isoclinic class of #20158
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	#3399	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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#20265, #6359626, #6359653, …
maximal normal subgroup of	C₂×#20265, #6359626, #6359653, …
maximal quotient of	#220897, #221013, #222427, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13253);