#94737

order	512
nilpotency class	6
isoclinism	isoclinic class of #94737
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7444
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₄×C₃₂	C₂²
prime radical	C₂×D₃₂	C₄
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

82 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=fgi,d^2=efg,e^2=gh,f^2=ghi,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,h^a=hi,c^b=cf,d^b=defg,e^b=eg,f^b=fgi,g^b=gh,h^b=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1933);