#106893

order	512
nilpotency class	3
isoclinism	isoclinic class of #106878
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#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

797 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:

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

Polycyclic presentation:

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

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, 14089);