#25363

order	256
nilpotency class	3
isoclinism	isoclinic class of #25140
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	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

358 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#25363, #6447131, #6447133, …
maximal normal subgroup of	C₂×#25363, #6447131, #6447133, …
maximal quotient of	#287200, #287206, #287212, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 18351);