#6498079

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1257 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)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,28)(29,33)(30,40)(31,35)(32,37)(34,38)(36,39);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,30,37,31,36,38,33)(26,32,34,28,35,29,40,39);(2,5)(6,8)(11,17)(14,21)(18,23)(22,24)(25,28,31,39)(26,27,35,36)(29,38,32,30)(33,34,37,40);(1,3,4,7)(2,6,5,8)(11,17)(14,21)(18,23)(22,24)(25,29)(26,33)(27,34)(28,38)(30,39)(31,32)(35,37)(36,40);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,36)(28,39)(29,32)(33,37)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6405275);