#6441858

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	#35252	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2150 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,19)(20,27,31,24)(21,25,32,28)(23,26,30,29);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,22,18)(20,28,31,25)(21,24,32,27)(23,26,30,29);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,20,22,31)(18,24,19,27)(21,30,32,23)(25,26,28,29);(1,4,5,3)(2,6,8,7)(17,21)(18,25)(19,28)(20,30)(22,32)(23,31)(24,26)(27,29);(18,19)(24,27)(25,28)(26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349054);