#6350440

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350397
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2930 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257636);