#199464

order	512
nilpotency class	3
isoclinism	isoclinic class of #199203
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1221 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi,d^c=dh,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, 106660);