#24930

order	256
nilpotency class	3
isoclinism	isoclinic class of #23886
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	#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

431 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)(3,5)(4,6)(7,8)(9,10)(11,20)(12,25)(13,15)(14,19)(16,24)(17,26)(18,29)(21,30)(22,37)(23,32)(27,34)(28,33)(31,39)(35,36)(38,40);(1,3,4,7)(2,5,6,8)(10,15)(11,21)(13,29)(17,25)(18,19)(22,36)(23,24)(26,28)(30,32)(33,34)(37,40)(38,39);(2,6)(5,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,30)(17,31)(18,32)(19,20)(21,29)(25,40)(26,36)(27,38)(28,39)(33,35)(34,37);(9,12,14,27)(10,17,19,33)(11,22,24,38)(13,28,29,26)(15,25,18,34)(16,31,20,35)(21,36,23,39)(30,40,32,37)

Polycyclic presentation:

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

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#24930, #6438378, #6438380, …
maximal normal subgroup of	C₂×#24930, #6438378, #6438380, …
maximal quotient of	#279836, #279837, #279848, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 17918);