#109321

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	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

493 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,e^2=h,f^2=hi,g^2=h,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, 16517);