#6674718

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(1,2)(3,4)(5,6)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19);(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(21,22,26,31)(23,32,34,42)(24,28,38,47)(25,29,40,30)(27,39,43,35)(33,48,36,50)(37,45,52,46)(41,51,44,49);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,23,26,34)(22,27,31,43)(24,36,38,33)(25,39,40,35)(28,44,47,41)(29,32,30,42)(37,49,52,51)(45,50,46,48);(6,13)(11,15)(12,14)(19,20)(21,24,26,38)(22,28,31,47)(23,33,34,36)(25,37,40,52)(27,41,43,44)(29,45,30,46)(32,48,42,50)(35,49,39,51);(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(22,29)(23,34)(25,40)(27,42)(28,45)(30,31)(32,43)(33,36)(37,52)(41,50)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6581914);