#109503

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=ci,d^b=dh,e^b=eh,d^c=dh

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, 16699);