#7168772

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

1160 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)(25,26,30,28)(27,34,37,40)(29,33,31,38)(32,39,35,36);(1,3,5,8)(2,6,4,7)(11,17)(12,20)(14,21)(15,19)(26,28)(32,35)(33,38)(34,40);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,30,37)(26,32,28,35)(29,36,31,39)(33,40,38,34);(1,4,5,2)(3,6,8,7)(11,17)(14,21)(18,23)(22,24)(25,28,30,26)(27,35,37,32)(29,33,31,38)(34,36,40,39);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,30,31)(26,33,28,38)(27,36,37,39)(32,40,35,34)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=i,d^2=f,e^2=hi,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, 7075968);