#116003

order	512
nilpotency class	3
isoclinism	isoclinic class of #116001
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eh,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23199);