#7168781

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 7075977);