#33728

order	256
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂²	C₂²×D₈
prime radical	C₂³	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,20)(12,17)(13,27)(14,29)(15,19)(16,24)(18,22)(21,33)(23,34)(25,39)(26,32)(28,40)(30,36)(31,38)(35,37);(1,3,4,8)(2,5,6,7)(11,21)(16,30)(20,33)(23,25)(24,36)(31,35)(34,39)(37,38);(9,11,22,37,14,25,28,36,15,21,26,38,12,23,13,24)(10,16,27,34,17,31,32,33,19,30,40,39,29,35,18,20);(9,12,15,14)(10,17,19,29)(11,23,21,25)(13,26,28,22)(16,31,30,35)(18,27,32,40)(20,34,33,39)(24,38,36,37)

Polycyclic presentation:

2^8|a^2=e,b^2=e,c^2=fg,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,c^b=ch

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#137527, #144016, #144020, …
maximal normal subgroup of	#137527, #144016, #144020, …
maximal quotient of	#494159, #495024, #495373, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26716);