#6355329

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355299
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60561	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1728 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,22,25)(19,21,24,31)(20,27,30,26)(23,28,32,29);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19,21,25,22,24,31,18)(20,26,28,32,30,27,29,23);(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,32)(29,31);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,21,22,31)(18,24,25,19)(20,28,30,29)(23,27,32,26);(17,21,22,31)(18,19,25,24)(20,28,30,29)(23,26,32,27)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262525);