#476123

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ei,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,d^c=de,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, 383319);