#7181992

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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	#2006	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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=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, 7089188);