#6571113

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1391 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fg,c^2=fg,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478309);