#31053

order	256
nilpotency class	3
isoclinism	isoclinic class of #30490
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

428 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)(3,7)(4,6)(5,8)(9,10,13,18,14,19,29,36)(11,15,21,17,23,32,22,31)(12,24,26,34,27,35,28,16)(20,37,38,40,39,33,25,30);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,25)(13,21)(14,23)(16,33)(17,18)(19,32)(20,26)(22,29)(24,30)(27,38)(28,39)(31,36)(34,37)(35,40);(9,12,14,27)(10,16,19,34)(11,20,23,39)(13,28,29,26)(15,30,32,40)(17,33,31,37)(18,35,36,24)(21,25,22,38);(2,6)(3,8)(9,11)(10,17)(12,20)(13,22)(14,23)(15,18)(16,33)(19,31)(21,29)(24,40)(25,26)(27,39)(28,38)(30,35)(32,36)(34,37)

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6667763, #6667765, #6667769, …
maximal normal subgroup of	#6667763, #6667765, #6667769, …
maximal quotient of	#342239, #342240, #342241, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 24041);