#6510905

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500370
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1415 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6418101);