#153608

order	512
nilpotency class	6
isoclinism	isoclinic class of #153607
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13661
commutator, Frattini, cosocle	C₂×C₃₂	C₂³
prime radical	#33981	C₂
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

176 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=df,e^b=ei,f^b=fg,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321773, #72321790, #72321829
maximal normal subgroup of	#72321773, #72321790, #72321829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60804);