#7065212

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
prime radical	#61169	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=hi,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 6972408);