#7145825

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996395
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

2114 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)(9,10,14,19,15,20,13,18)(11,21,23,31,25,36,24,16)(12,17,28,34,29,35,27,33)(22,32,26,39,38,40,37,30);(2,5)(3,7)(9,11)(10,16)(12,26)(13,23)(14,24)(15,25)(17,32)(18,21)(19,36)(20,31)(22,28)(27,38)(29,37)(30,34)(33,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,23)(22,30)(24,36)(25,31)(26,32)(27,33)(28,34)(29,35)(37,40)(38,39);(2,5)(6,8)(9,12,13,27,15,29,14,28)(10,17,18,33,20,35,19,34)(11,22,24,37,25,38,23,26)(16,30,36,40,31,39,21,32);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,21,31,36)(17,33,35,34)(22,26,38,37)(30,32,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7053021);