#444389

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

995 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:

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

Polycyclic presentation:

2^9|b^2=efi,c^2=g,d^2=ef,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,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, 351585);