#7148515

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^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, 7055711);