#6555569

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1620 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6462765);