#6686530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 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)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,26,30)(23,31,35,45)(24,28,36,42)(25,38,32,29)(27,39,41,34)(33,44,47,52)(37,49,46,43)(40,50,51,48);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(24,36)(28,42)(33,47)(37,46)(40,51)(43,49)(44,52)(48,50);(2,4)(21,23,32,39,26,35,25,34)(22,27,38,45,30,41,29,31)(24,33,46,50,36,47,37,48)(28,40,49,52,42,51,43,44);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24)(22,28)(23,33)(25,37)(26,36)(27,40)(29,43)(30,42)(31,44)(32,46)(34,48)(35,47)(38,49)(39,50)(41,51)(45,52);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(24,36)(28,42)(33,47)(37,46)(40,51)(43,49)(44,52)(48,50)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=f,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^d=ei

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, 6593726);