#6675186

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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

1796 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)(23,28)(24,35)(25,30)(26,39)(27,31)(29,36)(32,47)(33,42)(34,44)(37,45)(38,43)(40,46)(41,50)(48,51)(49,52);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(21,23,27,34)(22,28,31,44)(24,32,37,49)(25,38,26,33)(29,41,46,51)(30,43,39,42)(35,47,45,52)(36,50,40,48);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24,27,37)(22,29,31,46)(23,32,34,49)(25,36,26,40)(28,41,44,51)(30,45,39,35)(33,48,38,50)(42,47,43,52);(6,13)(11,15)(12,14)(19,20)(21,25,27,26)(22,30,31,39)(23,33,34,38)(24,36,37,40)(28,42,44,43)(29,45,46,35)(32,48,49,50)(41,47,51,52);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(24,37)(29,46)(32,49)(35,45)(36,40)(41,51)(47,52)(48,50)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582382);