#6353775

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353765
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	#60940	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1916 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)(9,10,15,18)(11,13,17,14)(12,16,20,23)(19,21,24,22);(2,5)(6,7)(9,11,14,10,15,17,13,18)(12,19,22,16,20,24,21,23);(2,6)(4,8)(5,7)(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,24)(18,23);(12,20)(16,23)(19,24)(21,22);(9,13,15,14)(10,17,18,11)(12,21,20,22)(16,24,23,19)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260971);