#6487818

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1880 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)(4,7)(5,8)(9,10,15,13)(11,19,22,37)(12,23,28,35)(14,30,31,18)(16,32,29,21)(17,26,25,27)(20,33,24,34)(36,38,40,39);(3,4)(6,7)(9,11,14,21,15,22,31,32)(10,16,18,19,13,29,30,37)(12,24,26,39,28,20,27,38)(17,34,35,40,25,33,23,36);(1,3,5,4)(2,6,8,7)(9,12,15,28)(10,17,13,25)(11,20,22,24)(14,26,31,27)(16,33,29,34)(18,35,30,23)(19,36,37,40)(21,38,32,39);(1,4,5,3)(2,7,8,6)(9,13,15,10)(11,16,22,29)(12,25,28,17)(14,30,31,18)(19,32,37,21)(20,33,24,34)(23,27,35,26)(36,39,40,38);(2,8)(6,7)(10,13)(16,29)(17,25)(18,30)(19,37)(23,35)(33,34)(36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,g^c=gi,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, 6395014);