#6443741

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439029
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60456	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2216 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,25)(13,18)(14,19)(15,21)(17,24)(20,30)(22,31)(23,32)(26,34)(27,35)(28,36)(29,37)(33,39)(38,40);(2,5)(6,8)(9,11)(10,16)(12,14)(13,26)(15,24)(17,21)(18,34)(19,25)(20,32)(22,28)(23,30)(27,39)(29,38)(31,36)(33,35)(37,40);(9,12,14,24,15,23,30,11)(10,17,20,25,21,16,19,32)(13,26,27,39,29,38,28,22)(18,33,36,34,37,31,35,40);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,22)(12,26)(14,27)(15,29)(16,31)(17,33)(19,35)(20,36)(21,37)(23,38)(24,39)(25,34)(28,30)(32,40);(9,14,15,30)(10,19,21,20)(11,23,24,12)(13,27,29,28)(16,32,17,25)(18,35,37,36)(22,38,39,26)(31,40,33,34)

Polycyclic presentation:

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

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, 6350937);