#6459598

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
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	#60444	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6366794);