#7149017

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
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
prime radical	#60837	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1785 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)(9,10,15,20,16,21,14,31)(11,22,25,13,19,30,24,17)(12,26,28,36,29,35,32,18)(23,33,38,40,34,27,39,37);(2,5)(3,7)(9,11,15,25,16,19,14,24)(10,17,31,30,21,13,20,22)(12,23,32,39,29,34,28,38)(18,33,26,40,36,27,35,37);(1,2)(3,6)(4,5)(7,8)(9,12,16,29)(10,18,21,36)(11,23,19,34)(13,27,17,33)(14,28,15,32)(20,35,31,26)(22,37,30,40)(24,38,25,39);(2,5)(6,8)(9,13)(10,19)(11,21)(12,27)(14,22)(15,30)(16,17)(18,34)(20,24)(23,36)(25,31)(26,39)(28,37)(29,33)(32,40)(35,38);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 7056213);