#7318

order	256
nilpotency class	4
isoclinism	isoclinic class of #7318
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1304
commutator	C₂²×C₄	C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

242 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#7318⋊C₃, #72266636
Fitting subgroup of	#7318⋊C₃
maximal subgroup of	#94622, #94623, #94698, …
maximal normal subgroup of	#94622, #94623, #94698, …
maximal quotient of	#93119, #93120, #93121, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 306);