#6467261

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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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)(11,20)(12,17)(13,29)(14,18)(15,19)(16,22)(21,32)(23,31)(24,35)(25,38)(26,33)(27,39)(28,34)(30,37)(36,40);(9,11,15,24)(10,16,19,31)(12,25,26,37)(13,23,14,22)(17,32,33,40)(18,35,29,20)(21,27,36,28)(30,39,38,34);(2,5)(6,8)(9,12,13,27,15,26,14,28)(10,17,29,39,19,33,18,34)(11,21,22,37,24,36,23,25)(16,30,35,40,31,38,20,32);(9,12,13,27,15,26,14,28)(10,17,29,39,19,33,18,34)(11,21,22,37,24,36,23,25)(16,30,35,40,31,38,20,32);(1,3,4,7)(2,6,5,8)(12,26)(17,33)(21,36)(25,37)(27,28)(30,38)(32,40)(34,39)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=hi,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,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, 6374457);