#114692

order	512
nilpotency class	3
isoclinism	isoclinic class of #114677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21888);