#28937

order	256
nilpotency class	3
isoclinism	isoclinic class of #25440
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	#3062	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

385 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)(11,21)(12,25)(13,16)(14,19)(15,20)(17,22)(18,26)(23,37)(24,30)(27,34)(28,31)(29,32)(33,38)(35,36)(39,40);(1,3,5,4)(2,6,7,8)(10,16,20,19)(11,22,24,29)(17,32)(18,31,34,25)(21,30)(23,36,38,39)(33,37)(35,40);(1,4,5,3)(2,6,7,8)(9,11,13,22,15,24,14,29)(10,17,19,21,20,32,16,30)(12,23,26,36,28,38,27,39)(18,33,25,35,34,37,31,40);(2,7)(6,8)(9,12,13,26,15,28,14,27)(10,18,19,25,20,34,16,31)(11,23,22,36,24,38,29,39)(17,33,21,35,32,37,30,40)

Polycyclic presentation:

2^8|b^2=eg,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^c=gh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#28937, #6460042, #6460047, …
maximal normal subgroup of	C₂×#28937, #6460042, #6460047, …
maximal quotient of	#325959, #325960, #325961, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 21925);