#444399

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

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

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=f,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, 351595);