#6563395

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1483 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)(11,17)(14,18)(19,23)(21,24)(25,26,31,40,30,34,29,35)(27,36,39,32,28,38,37,33);(1,3,5,4)(2,6,8,7)(9,10,13,16)(11,18,17,14)(12,15,20,22)(19,24,23,21)(25,27,30,28)(26,32,34,33)(29,39,31,37)(35,38,40,36);(1,3,5,4)(2,7,8,6)(9,11,13,17)(10,14,16,18)(12,19,20,23)(15,21,22,24)(25,28,30,27)(26,33,34,32)(29,37,31,39)(35,36,40,38);(1,4,5,3)(2,6,8,7)(12,20)(15,22)(19,23)(21,24);(9,12,13,20)(10,15,16,22)(11,19,17,23)(14,21,18,24)(26,34)(32,33)(35,40)(36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6470591);