#200557

order	512
nilpotency class	3
isoclinism	isoclinic class of #200412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1361 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107753);