#7547

order	256
nilpotency class	6
isoclinism	isoclinic class of D₆₄
automorphism	Aut(G) = C₂×#72224662 (order 4096) · Inn(G) = D₃₂ · Out(G) = C₂²×C₁₆
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	D₃₂
commutator	C₃₂	C₂×C₄
Frattini, cosocle	C₂×C₃₂	C₂²
socle	C₂	#1423
prime radical	C₂²	Q₆₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 128 points:

(1,2,5,12,9,15,8,38)(3,16,19,53,23,56,22,74)(4,24,26,75,29,78,28,84)(6,32,30,85,35,87,34,88)(7,36,31,51,37,90,33,14)(10,39,40,65,43,64,91,18)(11,44,45,61,47,60,92,17)(13,49,48,80,50,79,86,25)(20,68,66,117,71,119,70,120)(21,72,67,107,73,121,69,55)(27,82,81,124,83,125,58,77)(41,97,96,127,98,128,103,109)(42,63,89,95,99,116,126,108)(46,59,76,100,101,113,123,112)(52,94,102,111,104,110,122,57)(54,93,105,115,106,114,118,62);(1,3,17,57,6,20,58,108,33,69,112,128,79,114,28,64,8,22,60,110,34,70,83,116,37,73,113,127,80,115,29,65,9,23,61,111,35,71,81,95,31,67,100,97,49,93,26,39,5,19,44,94,30,66,27,63,7,21,59,109,25,62,4,18)(2,10,24,54,13,41,76,72,36,89,82,117,85,102,45,53,12,40,75,105,48,96,101,107,51,99,124,119,87,104,47,56,15,43,78,106,50,98,123,121,90,126,125,120,88,122,92,74,38,91,84,118,86,103,46,55,14,42,77,68,32,52,11,16)

Polycyclic presentation:

2^8|a^2=d,b^2=ce,c^2=ef,d^2=h,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94834, #94836, #94837, …
maximal normal subgroup of	#94834, #94836, #94837, …
maximal quotient of	#94749, #94750, #94762, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 535);