#6355400

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355367
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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1656 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)(3,9,10,16)(4,12,13,7)(6,14,15,11)(18,22)(19,25)(20,28)(24,30)(26,27)(29,31);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,18,20,22,21,24,28,30)(19,23,26,29,27,32,25,31);(17,19)(18,23)(20,26)(21,27)(22,29)(24,32)(25,28)(30,31);(3,10)(6,15)(9,16)(11,14);(17,20,21,28)(18,22,24,30)(19,26,27,25)(23,29,32,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,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=dhi,g^c=gi

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, 6262596);