#6495197

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35278	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1672 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:

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

Polycyclic presentation:

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

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, 6402393);