#6571532

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1328 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)(10,14)(15,17)(16,21)(22,23)(25,26,28,35,31,33,30,36)(27,34,37,40,29,39,38,32);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,31,29)(26,32,33,40)(28,38,30,37)(34,35,39,36);(1,3,5,4)(2,7,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,33)(27,37)(29,38)(30,31)(34,39);(1,4,5,3)(2,6,8,7)(12,20)(16,21)(18,24)(22,23)(25,29,31,27)(26,34,33,39)(28,38,30,37)(32,35,40,36);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(27,29)(32,40)(34,39)(37,38)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fgi,c^2=fg,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ei,f^b=fi,g^b=gi,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, 6478728);