#109326

order	512
nilpotency class	3
isoclinism	isoclinic class of #109288
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₄⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,8,6,9,5,14)(3,10,12,16,11,7,13,15)(17,18,21,28,23,26,22,27)(19,29,32,42,33,24,31,39)(20,34,35,25,37,41,36,40)(30,44,46,38,47,48,45,43);(1,3,5,13,6,11,4,12)(2,7,8,15,9,10,14,16)(17,19,22,32,23,33,21,31)(18,24,27,39,26,29,28,42)(20,30,36,46,37,47,35,45)(25,38,34,48,40,43,41,44);(17,20,21,35,23,37,22,36)(18,25,27,34,26,40,28,41)(19,30,31,45,33,47,32,46)(24,38,39,48,29,43,42,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,e^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,f^b=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16522);