#6467255

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442587
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	#60565	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1869 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)(11,16)(12,17)(13,15)(14,19)(18,26)(20,27)(21,30)(22,29)(23,28)(24,33)(25,32)(31,36)(34,37)(35,39)(38,40);(10,15)(11,20)(12,23)(13,26)(16,29)(17,32)(18,19)(21,24)(22,36)(25,37)(30,40)(33,39);(2,5)(6,8)(9,11,13,20,14,22,26,36)(10,16,15,27,19,29,18,31)(12,24,23,38,25,21,37,35)(17,33,28,40,32,30,34,39);(9,12,13,23,14,25,26,37)(10,17,15,28,19,32,18,34)(11,21,20,35,22,24,36,38)(16,30,27,39,29,33,31,40);(1,3,4,7)(2,6,5,8)(12,25)(17,32)(21,24)(23,37)(28,34)(30,33)(35,38)(39,40)

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^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, 6374451);