#153425

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₃₂	C₄²
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₃₂	C₂³
socle	C₂	C₂×#1397
prime radical	C₂³	M₆₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

134 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bh,e^a=ei,c^b=ci,d^b=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321329, #72321360, #72321369, …
maximal normal subgroup of	#72321329, #72321360, #72321369, …
maximal quotient of	#72234398

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60621);