#6447323

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1782 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,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, 6354519);