#200587

order	512
nilpotency class	3
isoclinism	isoclinic class of #200424
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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

1259 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:

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

Polycyclic presentation:

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

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, 107783);