#110093

order	512
nilpotency class	3
isoclinism	isoclinic class of #110086
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17289);