#431921

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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

750 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=f,d^2=efi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi

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, 339117);