#151751

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₃₂	C₄²
commutator	C₄	C₄²×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12539
prime radical	C₂³	#504
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 128 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,g^2=h,h^2=i,c^b=ch,f^b=fi,e^c=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72234252

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58947);