#153581

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

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

111 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 136 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60777);