#6351636

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351632
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
prime radical	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3065 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)(9,10,15,12)(11,19,22,24)(13,18,14,17)(16,23,20,21);(3,7)(5,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,19)(18,24);(1,2)(3,8)(4,6)(5,7)(9,12,15,10)(11,20,22,16)(13,17,14,18)(19,23,24,21);(9,13,15,14)(10,17,12,18)(11,21,22,23)(16,19,20,24);(11,22)(16,20)(19,24)(21,23)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258832);