#6677361

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

1745 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)(21,22,26,27)(23,31,34,28)(24,29,37,41)(25,38,40,30)(32,35,48,42)(33,43,50,46)(36,45,39,44)(47,52,49,51);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23)(22,28)(24,35)(25,33)(26,34)(27,31)(29,32)(30,43)(36,51)(37,42)(38,46)(39,52)(40,50)(41,48)(44,47)(45,49);(2,4)(21,24,26,37)(22,29,27,41)(23,32,34,48)(25,39,40,36)(28,42,31,35)(30,45,38,44)(33,49,50,47)(43,51,46,52);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,26,40)(22,30,27,38)(23,33,34,50)(24,36,37,39)(28,43,31,46)(29,44,41,45)(32,47,48,49)(35,51,42,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(22,27)(23,31)(28,34)(29,41)(30,38)(32,35)(33,46)(42,48)(43,50)(44,45)(47,51)(49,52)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6584557);