#6484542

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470092
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6391738);