#6467262

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442584
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	#60559	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1901 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,18)(13,19)(14,31)(15,20)(16,21)(17,25)(23,35)(24,32)(26,34)(27,38)(28,40)(29,36)(30,37)(33,39);(9,11)(10,17)(12,28)(13,23)(14,26)(15,25)(16,27)(18,35)(19,32)(20,38)(21,34)(22,31)(24,29)(30,39)(33,36)(37,40);(2,5)(6,8)(9,12,15,13,16,29,14,30)(10,18,20,19,21,36,31,37)(11,23,26,28,27,39,25,24)(17,32,22,35,34,40,38,33);(9,13,15,29,16,30,14,12)(10,19,20,36,21,37,31,18)(11,24,26,23,27,28,25,39)(17,33,22,32,34,35,38,40);(1,3,4,7)(2,6,5,8)(12,29)(13,30)(18,36)(19,37)(23,39)(24,28)(32,40)(33,35)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374458);