#8789

order	256
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=f,b^2=e,c^2=d,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cgh

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#93737, #99414, #99415, …
maximal normal subgroup of	#93737, #99414, #99415, …
maximal quotient of	#99289, #108516, #108786, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1777);