#6350416

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
prime radical	C₂×H(5,𝔽₂)	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2769 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^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, 6257612);