#6441339

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1730 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)(11,13)(12,14)(15,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,31)(28,32)(29,30);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,28)(22,29)(24,31)(26,32);(10,14)(13,16)(17,19,21,20)(18,24,23,26)(22,28,29,27)(25,31,30,32);(1,4,5,3)(2,6,7,8)(17,20,21,19)(18,24,23,26)(22,27,29,28)(25,31,30,32);(18,23)(24,26)(25,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348535);