#26848

order	256
nilpotency class	3
isoclinism	isoclinic class of #24289
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	C₂²×C₄	C₂⁴
prime radical	#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

359 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#26848, #6443957, #6443963, …
maximal normal subgroup of	C₂×#26848, #6443957, #6443963, …
maximal quotient of	#311001, #311004, #311006, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 19836);