#6672952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61174	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1923 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,24)(13,30)(14,18)(15,19)(17,27)(20,37)(21,31)(22,33)(23,34)(25,32)(26,28)(29,36)(35,38)(39,40);(2,5)(3,7)(9,11,15,23)(10,16,19,34)(12,25,29,39)(13,22,14,31)(17,35,26,37)(18,21,30,33)(20,27,38,28)(24,32,36,40);(9,12,15,29)(10,17,19,26)(11,20,23,38)(13,28,14,27)(16,32,34,40)(18,36,30,24)(21,35,33,37)(22,39,31,25);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,10)(11,21)(12,26)(13,18)(14,30)(15,19)(16,31)(17,29)(20,37)(22,34)(23,33)(24,28)(25,40)(27,36)(32,39)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6580148);