#6675125

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1787 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582321);