#6473173

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473167
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	#60755	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1964 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)(3,5)(4,6)(7,8)(9,10,15,18)(11,19,24,20)(12,14,26,30)(13,17,28,33)(16,23,32,22)(21,34,36,35)(25,29,27,39)(31,38,40,37);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,24)(17,31)(18,32)(19,26)(22,30)(25,35)(27,34)(28,36)(29,38)(33,40)(37,39);(3,7)(5,8)(9,12,14,18,15,26,30,10)(11,20,23,32,24,19,22,16)(13,27,29,17,28,25,39,33)(21,34,38,31,36,35,37,40);(3,7)(5,8)(9,13)(10,17)(11,21)(12,25)(14,29)(15,28)(16,31)(18,33)(19,34)(20,35)(22,37)(23,38)(24,36)(26,27)(30,39)(32,40);(2,6)(5,8)(13,28)(17,33)(21,36)(25,27)(29,39)(31,40)(34,35)(37,38)

Polycyclic presentation:

2^9|a^2=i,c^2=g,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,f^b=fi,d^c=di,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, 6380369);