#94762

order	512
nilpotency class	6
isoclinism	isoclinic class of D₆₄
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₃₂
commutator	C₃₂	C₄²
Frattini, cosocle	C₂²×C₃₂	C₂²
socle	C₂²	#1423
prime radical	C₂³	Q₆₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 132 points:

(1,2)(3,4)(5,6,9,16,13,21,36,20)(7,22,24,67,28,70,77,69)(8,17,30,55,32,61,87,60)(10,15,33,51,39,54,91,53)(11,40,34,92,41,97,85,96)(12,42,35,19,43,57,93,65)(14,44,45,73,50,100,49,23)(18,62,56,86,64,115,63,29)(25,66,74,116,80,118,123,117)(26,81,75,124,82,129,121,128)(27,83,76,68,84,119,125,120)(31,88,37,59,89,113,94,114)(38,58,90,52,95,111,130,112)(46,99,101,122,105,131,104,71)(47,106,102,127,108,132,107,79)(48,109,103,126,110,72,98,78);(2,4)(5,7,10,25,11,26,37,78,35,76,90,106,62,99,30,44,9,24,33,74,34,75,89,109,43,84,95,127,86,122,32,73,13,28,39,80,41,82,94,126,93,125,130,132,115,131,87,100,36,77,91,123,85,121,31,72,12,27,38,79,29,71,8,23)(6,14,17,46,18,47,58,83,42,98,88,128,96,117,53,69,20,49,60,104,63,107,112,120,65,110,114,129,97,118,54,70,21,50,61,105,64,108,111,119,57,103,113,124,92,116,51,67,16,45,55,101,56,102,52,68,19,48,59,81,40,66,15,22)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=i,e^2=f,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fg,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, 1958);