#6587590

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1438 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,23,22)(19,24,21,26)(20,25,30,32)(27,29,31,28);(1,3,5,4)(2,6,7,8)(17,19,23,21)(18,24,22,26)(20,28,30,29)(25,27,32,31);(10,13)(14,16)(17,20,23,30)(18,25,22,32)(19,27,21,31)(24,29,26,28);(9,11,12,15)(10,14,13,16)(17,21,23,19)(18,26,22,24)(20,29,30,28)(25,31,32,27);(1,4,5,3)(2,6,7,8)(18,22)(19,26)(20,30)(21,24)(27,29)(28,31)

Polycyclic presentation:

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

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, 6494786);