#115978

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

788 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:

(1,2,3,5,4,6,7,8)(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);(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,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eh,g^c=gh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23174);