#6686625

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1693 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:

(2,4)(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,25,29)(23,31,35,27)(24,36,37,28)(26,30,39,44)(32,42,34,40)(33,41,47,45)(38,49,50,43)(46,51,48,52);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(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);(1,2,3,4)(23,32)(26,39)(27,40)(30,44)(31,42)(33,46)(34,35)(38,50)(41,51)(43,49)(45,52)(47,48);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(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)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,34)(24,37)(26,39)(27,42)(28,36)(30,44)(31,40)(32,35)(33,46)(41,51)(45,52)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6593821);