#6677547

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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1741 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(22,27)(24,36)(28,40)(30,31)(32,45)(33,47)(37,49)(48,50);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,22,25,27)(23,32,34,45)(24,29,36,41)(26,30,38,31)(28,39,40,35)(33,46,47,52)(37,44,49,43)(42,50,51,48);(2,4)(21,23,25,34)(22,28,27,40)(24,33,36,47)(26,39,38,35)(29,42,41,51)(30,32,31,45)(37,50,49,48)(43,52,44,46);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24)(22,29)(23,33)(25,36)(26,37)(27,41)(28,42)(30,44)(31,43)(32,46)(34,47)(35,48)(38,49)(39,50)(40,51)(45,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(22,30)(24,36)(26,38)(27,31)(28,32)(29,43)(33,47)(35,39)(40,45)(41,44)(42,52)(46,51)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,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, 6584743);