#7145816

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

2078 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)(9,10,15,19,14,20,13,18)(11,21,24,16,25,36,23,31)(12,17,29,34,28,35,27,33)(22,32,37,30,38,40,26,39);(2,5)(3,7)(9,11)(10,16)(12,26)(13,24)(14,25)(15,23)(17,32)(18,36)(19,21)(20,31)(22,27)(28,37)(29,38)(30,33)(34,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,23)(14,25)(15,24)(17,30)(18,21)(19,36)(20,31)(26,27)(28,38)(29,37)(32,33)(34,40)(35,39);(2,5)(6,8)(9,12,13,27,14,28,15,29)(10,17,18,33,20,35,19,34)(11,22,23,26,25,38,24,37)(16,30,21,32,31,39,36,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=fi,d^2=f,e^2=h,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, 7053012);