#338717

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=eh,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 245913);