#94751

order	512
nilpotency class	6
isoclinism	isoclinic class of D₆₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	D₃₂
commutator	C₃₂	C₂×C₈
Frattini, cosocle	C₄×C₃₂	C₂²
socle	C₂	#7460
prime radical	C₂²	#1425
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

47 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=ce,c^2=eg,d^2=f,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1947);