#443559

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

1770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=ei,c^2=eg,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,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, 350755);