#6350599

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
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

2826 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,22,26)(19,27,28,25)(20,23,21,24)(29,32,30,31);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(18,23)(20,21)(24,26)(25,31)(27,32)(29,30);(18,24)(20,21)(23,26)(25,32)(27,31)(29,30);(2,7)(6,15)(8,12)(9,16)(17,19)(18,25)(20,29)(21,30)(22,28)(23,31)(24,32)(26,27);(19,28)(25,27)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257795);