#6354208

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354207
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60968	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1826 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)(9,10,14,16)(11,13,18,23)(12,19,20,17)(15,21,24,22);(9,11,13,16,14,18,23,10)(12,17,21,24,20,19,22,15);(2,5)(3,8)(6,7)(9,12)(10,15)(11,17)(13,21)(14,20)(16,24)(18,19)(22,23);(2,6)(5,7)(10,16)(11,18)(15,24)(17,19);(12,20)(15,24)(17,19)(21,22)

Polycyclic presentation:

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

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, 6261404);