#6353770

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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1787 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^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, 6260966);