#6447292

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1816 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=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, 6354488);