#6351561

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351559
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3242 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257726
maximal normal subgroup of	#72257726
maximal quotient of	#72257861

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258757);