#10303

order	256
nilpotency class	3
isoclinism	isoclinic class of #10300
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

239 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^8|a^2=g,b^2=g,c^2=h,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,f^b=fgh,e^c=eh,f^c=fgh

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#10303⋊C₃
maximal subgroup of	#151365, C₂×#10303, #233322, …
maximal normal subgroup of	#151365, C₂×#10303, #233322, …
maximal quotient of	#102738, #102740, #102742, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 3291);