#6352345

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352341
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|d^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei

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, 6259541);