#6916735

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884809
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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6823931);