#200827

order	512
nilpotency class	3
isoclinism	isoclinic class of #200746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,7,10)(3,6,9,5)(4,11)(8,13)(12,16)(14,15)(17,18,22,27,23,26,21,34)(19,24,30,40,29,38,36,39)(20,31,33,25,32,42,35,41)(28,43,46,47,44,48,45,37);(1,3,5,2,7,9,6,10)(4,11,14,8,12,16,13,15)(17,19,21,30,23,29,22,36)(18,24,27,39,26,38,34,40)(20,28,35,46,32,44,33,45)(25,37,42,48,41,47,31,43);(1,4)(2,8)(3,11)(5,14)(6,13)(7,12)(9,16)(10,15)(17,20,23,32)(18,25,26,41)(19,28,29,44)(21,35,22,33)(24,37,38,47)(27,42,34,31)(30,46,36,45)(39,48,40,43);(4,12)(8,15)(11,16)(13,14)(18,26)(20,32)(24,38)(27,34)(28,44)(33,35)(39,40)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108023);