#6526555

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500385
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1748 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:

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

Polycyclic presentation:

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

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, 6433751);