#6687385

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,18,22,25)(19,23,26,20)(21,27,29,32)(24,28,31,30);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,19,20,25,22,26,23,18)(21,27,28,31,29,32,30,24);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,16,11,15)(17,20,22,23)(18,19,25,26)(21,28,29,30)(24,27,31,32);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=f,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

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, 6594581);