#94576

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

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

77 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 136
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 136 points:

(1,2)(3,7,8,5)(4,6)(9,10,13,20,14,21,36,55,15,22,37,56,40,59,83,104,16,23,38,57,41,60,84,105,43,62,86,107,88,109,124,132,17,24,39,58,42,61,85,106,44,63,87,108,89,110,125,133,12,19,32,51,33,52,77,99,34,53,78,100,79,101,120,130)(11,25,27,64,28,65,70,111,29,66,71,112,72,113,117,134,26,49,67,54,68,95,114,102,69,96,115,103,116,128,135,131,30,50,35,80,73,97,81,121,74,98,82,122,118,129,123,136,31,18,45,46,75,47,90,92,76,48,91,93,119,94,126,127);(1,3,4,8)(2,5,6,7)(9,11)(10,18)(12,31)(13,35)(14,28)(15,29)(16,26)(17,30)(19,50)(20,54)(21,47)(22,48)(23,25)(24,49)(27,39)(32,67)(33,75)(34,76)(36,81)(37,82)(38,45)(40,72)(41,68)(42,73)(43,69)(44,74)(46,58)(51,64)(52,97)(53,98)(55,102)(56,103)(57,80)(59,94)(60,65)(61,95)(62,66)(63,96)(70,85)(71,87)(77,114)(78,115)(79,119)(83,123)(84,90)(86,91)(88,116)(89,118)(92,106)(93,108)(99,111)(100,112)(101,129)(104,131)(105,121)(107,122)(109,113)(110,128)(117,125)(120,135)(124,126)(127,133)(130,134)(132,136)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ce,c^b=ci,d^b=dei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1772);