#6477758

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470415
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	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1834 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)(19,26)(20,30)(24,25)(28,29)(31,32);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,18,20,22,21,24,30,25)(19,27,28,32,29,23,26,31);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,19,21,29)(18,23,24,27)(20,28,30,26)(22,31,25,32);(3,10)(6,14)(9,11)(13,15)(18,24)(22,25)(23,27)(31,32);(2,8)(6,14)(7,16)(13,15)(17,20,21,30)(18,25,24,22)(19,28,29,26)(23,32,27,31)

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,e^b=ei,g^c=gi

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, 6384954);