#25271

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	#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

364 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)(3,6)(4,5)(7,8)(9,10,15,28,14,19,13,18)(11,16,29,40,22,32,21,31)(12,23,25,17,27,34,26,35)(20,36,37,30,24,39,38,33);(2,5)(6,8)(9,11,13,21,14,22,15,29)(10,16,18,31,19,32,28,40)(12,24,26,37,27,20,25,38)(17,33,23,39,35,30,34,36);(1,3,4,7)(2,6,5,8)(9,12,14,27)(10,17,19,35)(11,20,22,24)(13,25,15,26)(16,30,32,33)(18,34,28,23)(21,37,29,38)(31,39,40,36);(2,5)(6,8)(9,13,14,15)(10,18,19,28)(11,21,22,29)(12,25,27,26)(16,31,32,40)(17,34,35,23)(20,37,24,38)(30,39,33,36)

Polycyclic presentation:

2^8|a^2=fh,b^2=f,c^2=eh,d^2=h,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×#25271, #6441696, #6441698, …
maximal normal subgroup of	C₂×#25271, #6441696, #6441698, …
maximal quotient of	#285664, #285665, #285668, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 18259);