#6674297

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

1806 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)(21,22)(23,31)(24,36)(25,29)(26,30)(27,35)(28,37)(32,42)(33,41)(34,40)(38,49)(39,44)(43,50)(45,47)(46,51)(48,52);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,23,25,35)(22,27,29,31)(24,33,37,47)(26,32,39,34)(28,41,36,45)(30,40,44,42)(38,46,50,48)(43,51,49,52);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(23,32)(26,39)(27,40)(30,44)(31,42)(33,46)(34,35)(38,50)(41,51)(43,49)(45,52)(47,48);(6,13)(11,15)(12,14)(19,20)(21,24)(22,28)(23,33)(25,37)(26,38)(27,41)(29,36)(30,43)(31,45)(32,46)(34,48)(35,47)(39,50)(40,51)(42,52)(44,49);(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(22,29)(23,34)(24,37)(26,39)(27,40)(31,42)(32,35)(33,46)(41,52)(43,49)(45,51)(47,48)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6581493);