#6555869

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#459	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=i,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ei,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, 6463065);