#94763

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	#1415
prime radical	C₂³	SD₆₄
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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,e^2=fi,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, 1959);