#6476545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470368
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1620 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,23)(21,25)(22,26)(24,28)(29,32)(30,31);(9,11)(10,13)(12,15)(14,16)(17,19,22,28,23,29,21,31)(18,24,25,27,20,30,26,32);(2,5)(6,8)(11,15)(13,16)(17,20,23,18)(19,24,29,30)(21,25,22,26)(27,31,32,28);(11,15)(13,16)(17,21,23,22)(18,25,20,26)(19,28,29,31)(24,32,30,27);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,23,22)(18,26,20,25)(19,28,29,31)(24,27,30,32)

Polycyclic presentation:

2^9|b^2=f,c^2=i,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=dh,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6383741);