#6676529

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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	#30477	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1721 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=fi,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,g^c=gi,h^c=hi,e^d=ei,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, 6583725);