#6477754

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470409
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	C₂²×C₄	C₂⁵
prime radical	C₂×#3188	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1853 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,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, 6384950);