#6477249

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1608 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=g,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384445);