#7148513

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055709);