#6520047

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	#60919	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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,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, 6427243);