#152385

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 76
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 76 points:

(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,29)(16,22)(17,39)(18,43)(19,45)(20,48)(21,28)(23,34)(24,30)(25,42)(26,40)(27,46)(31,53)(32,64)(33,54)(35,59)(36,58)(37,70)(38,52)(41,56)(44,55)(47,74)(49,68)(50,67)(51,69)(57,73)(60,65)(61,72)(62,71)(63,76)(66,75);(6,9)(10,12)(13,15,30,36,19,32,61,69,46,65,44,63,73,68,42,38,21,34,18,31,47,66,41,37,20,33,62,67,40,35,17,16)(14,22,39,59,26,50,71,54,48,70,56,75,74,53,43,23,28,52,25,49,57,76,55,60,27,51,72,64,45,58,24,29);(5,7,8,11)(6,10,9,12)(13,16,17,35,40,67,62,33,20,37,41,66,47,31,18,34,21,38,42,68,73,63,44,65,46,69,61,32,19,36,30,15)(14,23,24,53,45,75,72,70,27,54,55,50,57,59,25,22,28,29,43,58,74,64,56,51,48,60,71,76,26,49,39,52)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59581);