#9006

order	256
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#120
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#386
prime radical	C₂³	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,7,10,6,11,5,9)(3,12,14,8,15,16,13,4)(17,18)(19,28,31,39)(20,32)(21,26)(22,36)(23,27)(24,30,40,35)(25,33)(29,44,47,38)(34,41)(37,42)(43,45,48,46);(1,3)(2,8)(4,11)(5,14)(6,15)(7,13)(9,16)(10,12)(17,19,21,35,23,31,22,30)(18,24,26,39,27,40,36,28)(20,29,33,46,34,47,37,45)(25,38,41,48,42,44,32,43);(1,4,6,8)(2,3,11,15)(5,12,7,16)(9,14,10,13)(17,20,23,34)(18,25,27,42)(19,29,31,47)(21,33,22,37)(24,38,40,44)(26,41,36,32)(28,43,39,48)(30,45,35,46)

Polycyclic presentation:

2^8|a^2=f,b^2=df,c^2=g,d^2=g,e^2=gh,f^2=h,b^a=bd,c^a=ce,e^a=eg,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,f^c=fh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#9006, #194610, #194612, …
maximal normal subgroup of	C₂×#9006, #194610, #194612, …
maximal quotient of	#99536, #99542, #99672, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1994);