#6355328

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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1706 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)(3,9,12,16)(4,13,10,7)(6,14,15,11)(18,23)(19,27)(20,21)(25,26)(28,29)(30,32);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(17,18,20,26,22,25,21,23)(19,24,27,32,29,31,28,30);(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32);(18,25)(23,26)(24,31)(30,32);(17,20,22,21)(18,26,25,23)(19,27,29,28)(24,32,31,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,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, 6262524);