#6520049

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502937
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	#60703	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1652 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6427245);