#339746

order	512
nilpotency class	3
isoclinism	isoclinic class of #339703
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25404	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 246942);