#7317

order	256
nilpotency class	4
isoclinism	isoclinic class of #7316
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1304
commutator, prime radical	C₂⁴	C₄²
Frattini, cosocle	C₂³×Q₈	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=d,b^2=e,d^2=h,e^2=h,b^a=bc,c^a=cf,e^a=eg,g^a=gh,c^b=cg,d^b=df,f^b=fh,e^d=eh

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#7317⋊C₃, C₂²×#7317⋊C₃, #72266652, …
Fitting subgroup of	#7317⋊C₃
maximal subgroup of	#94620, #94621, C₂×#7317, …
maximal normal subgroup of	#94620, #94621, C₂×#7317, …
maximal quotient of	#93106, #93107, #93108, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 305);