#6355611

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355609
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1789 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:

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

Polycyclic presentation:

2^9|b^2=fh,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,e^c=eh,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, 6262807);