#6473180

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	#60755	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1966 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)(9,10,16,19)(11,21,25,22)(12,15,20,14)(13,18,29,36)(17,32,34,24)(23,37,27,38)(26,31,28,30)(33,40,35,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,21)(13,27)(14,32)(15,24)(16,25)(18,35)(19,34)(20,22)(23,29)(26,38)(28,37)(30,39)(31,40)(33,36);(3,7)(5,8)(9,12,15,19,16,20,14,10)(11,22,24,17,25,21,32,34)(13,28,31,18,29,26,30,36)(23,37,39,35,27,38,40,33);(3,7)(5,8)(9,13)(10,18)(11,23)(12,26)(14,30)(15,31)(16,29)(17,33)(19,36)(20,28)(21,37)(22,38)(24,39)(25,27)(32,40)(34,35);(2,6)(5,8)(10,19)(12,20)(13,29)(17,34)(21,22)(23,27)(30,31)(39,40)

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^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, 6380376);