#6727905

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1259 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)(4,7)(5,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(28,36)(33,40)(35,38)(37,39);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,26,30,34)(27,32,29,31)(28,37,36,39)(33,38,40,35);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(26,31)(27,29)(32,34)(33,37)(35,38)(39,40);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,27,30,29)(26,32,34,31)(28,35,36,38)(33,39,40,37);(1,4,5,3)(2,7,8,6)(11,21)(15,18)(19,20)(23,24)(25,28,30,36)(26,33,34,40)(27,35,29,38)(31,37,32,39)

Polycyclic presentation:

2^9|b^2=gi,c^2=fg,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6635101);