#6494251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473174
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	#60920	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1718 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6401447);