#409127

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=f,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,g^b=gi,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, 316323);