#7143619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2141 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 7050815);