#153568

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

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

126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 132 points:

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

Polycyclic presentation:

2^9|a^2=e,b^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, 60764);