#6675387

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	#36019	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1765 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)(21,22,25,29)(23,31,34,27)(24,36,37,28)(26,30,39,44)(32,42,35,40)(33,41,48,45)(38,49,50,43)(46,51,47,52);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,25,34)(22,27,29,31)(24,33,37,48)(26,32,39,35)(28,41,36,45)(30,40,44,42)(38,46,50,47)(43,51,49,52);(2,4)(23,32)(24,37)(26,39)(27,40)(28,36)(30,44)(31,42)(33,47)(34,35)(41,52)(45,51)(46,48);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,25,37)(22,28,29,36)(23,33,34,48)(26,38,39,50)(27,41,31,45)(30,43,44,49)(32,46,35,47)(40,51,42,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,32)(26,39)(27,40)(30,44)(31,42)(33,46)(34,35)(38,50)(41,51)(43,49)(45,52)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^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, 6582583);