#6432811

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432811
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#401
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

1729 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340007);