#6369236

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369207
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276432);