#105343

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄²	C₂³
commutator	C₂³	C₄³
socle, prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ci,c^b=ch

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12539);