#94744

order	512
nilpotency class	6
isoclinism	isoclinic class of #94743
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7451
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₄×C₃₂	C₂²
prime radical	C₂²	#1419
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

59 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=eh,c^2=fgi,d^2=efgi,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=def,e^b=ei,f^b=fg,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, 1940);