#6587596

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1437 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)(19,22)(20,24)(21,28)(23,26)(25,32)(27,31)(29,30);(1,3,5,4)(2,6,7,8)(17,19,22,18)(20,25,31,28)(21,27,32,24)(23,26,30,29);(10,13)(14,16)(17,20,22,31)(18,24,19,27)(21,23,32,30)(25,29,28,26);(9,11,12,15)(10,14,13,16)(20,21)(23,30)(24,28)(25,27)(26,29)(31,32);(1,4,5,3)(2,6,7,8)(17,21,22,32)(18,25,19,28)(20,30,31,23)(24,26,27,29)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=g,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=ehi,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, 6494792);