#7173233

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1364 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,31,33)(27,35,29,28)(30,39,37,34)(32,38,40,36);(1,3,4,7)(2,5,6,8)(11,17)(12,20)(14,21)(15,19)(25,27,30,36,31,29,37,38)(26,32,34,35,33,40,39,28);(2,6)(5,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28,31,35)(26,29,33,27)(30,32,37,40)(34,38,39,36);(11,17)(14,21)(18,23)(22,24)(25,29,31,27)(26,28,33,35)(30,36,37,38)(32,34,40,39);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,33)(28,35)(32,40)(34,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

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, 7080429);