#6347338

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347305
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	C₂²×C₄	C₂⁵
prime radical	#34936	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

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

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, 6254534);