#6490728

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469995
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

1324 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,29,32,30,31,38,39)(27,33,35,36,34,40,37,28);(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27,30,34)(26,28,31,36)(29,37,38,35)(32,40,39,33);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(27,34)(28,36)(33,40)(35,37);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28)(26,27)(29,33)(30,36)(31,34)(32,35)(37,39)(38,40);(1,4,5,3)(2,6,7,8)(10,18)(15,24)(16,17)(22,23)(26,31)(28,36)(32,39)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6397924);