#6571210

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×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fg,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, 6478406);