#6917301

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

1398 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,d^2=f,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, 6824497);