#6629711

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×#3062	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2168 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)(12,22)(13,24)(16,25)(17,30)(18,27)(19,32)(21,34)(26,31)(28,38)(29,37)(36,39);(1,3,4,7)(2,5,6,8)(9,10,13,15,14,18,24,27)(11,16,21,25,20,29,34,37)(12,17,23,26,22,31,35,30)(19,28,33,36,32,39,40,38);(2,6)(5,8)(9,11)(10,16)(12,19)(13,21)(14,20)(15,25)(17,28)(18,29)(22,32)(23,33)(24,34)(26,36)(27,37)(30,38)(31,39)(35,40);(9,12,13,23,14,22,24,35)(10,17,15,26,18,31,27,30)(11,19,21,33,20,32,34,40)(16,28,25,36,29,39,37,38);(11,20)(16,29)(19,32)(21,34)(25,37)(28,39)(33,40)(36,38)

Polycyclic presentation:

2^9|b^2=fg,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,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, 6536907);