#6675391

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	#42650	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1769 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)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,26,24)(23,31,35,28)(25,37,39,29)(27,30,41,38)(32,36,43,34)(33,42,49,45)(40,51,52,44)(46,48,47,50);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23)(22,28)(24,31)(25,33)(26,35)(27,34)(29,42)(30,43)(32,38)(36,41)(37,45)(39,49)(40,48)(44,47)(46,51)(50,52);(2,4)(21,24,26,22)(23,32,35,43)(25,29,39,37)(27,30,41,38)(28,34,31,36)(33,47,49,46)(40,51,52,44)(42,50,45,48);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,26,39)(22,29,24,37)(23,33,35,49)(27,40,41,52)(28,42,31,45)(30,44,38,51)(32,46,43,47)(34,48,36,50);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,34)(27,41)(28,43)(30,38)(31,32)(33,48)(35,36)(40,52)(42,47)(44,51)(45,46)(49,50)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,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, 6582587);