#6677552

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1739 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)(21,22,27,31)(23,28,35,45)(24,29,39,48)(25,30,40,26)(32,43,50,51)(33,34,44,41)(36,46,49,52)(37,38,47,42);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,27,35)(22,28,31,45)(24,36,39,49)(25,33,40,44)(26,41,30,34)(29,46,48,52)(32,42,50,38)(37,51,47,43);(2,4)(21,24,27,39)(22,29,31,48)(23,32,35,50)(25,37,40,47)(26,42,30,38)(28,43,45,51)(33,46,44,52)(34,49,41,36);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25)(22,30)(23,33)(24,37)(26,31)(27,40)(28,34)(29,38)(32,46)(35,44)(36,51)(39,47)(41,45)(42,48)(43,49)(50,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,34)(25,40)(26,30)(28,44)(32,49)(33,45)(35,41)(36,50)(37,47)(38,42)(43,52)(46,51)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=f,e^2=f,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, 6584748);