#6629714

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629034
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=fgi,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,e^b=ei,e^c=ei,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536910);