#7330

order	256
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	Aut(G) = #72283979 (order 2048) · Inn(G) = C₄² · Out(G) = #3003
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₄²
commutator	C₄	C₈²
Frattini, cosocle	C₄×C₁₆	C₂²
socle, prime radical	C₂²	#346
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

55 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

2^8|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,g^2=h,b^a=bg,d^a=dh,c^b=ch

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	C₂×#72224644, C₂×#72224648, C₂×#72224654, …
maximal subgroup of	#93668, #93766, #93767, …
maximal normal subgroup of	#93668, #93766, #93767, …
maximal quotient of	#93666, #93990, #94506, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 318);