#200728

order	512
nilpotency class	3
isoclinism	isoclinic class of #200728
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

1073 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)(3,6,10,14)(4,7)(5,8)(9,12)(11,13,16,15)(17,18)(19,28)(20,32,35,42)(21,36)(22,26)(23,27)(24,30)(25,34,41,33)(29,44,46,38)(31,39)(37,40)(43,45,48,47);(1,3)(2,6,8,14)(4,11)(5,10)(7,13,9,15)(12,16)(17,19,21,30,23,31,22,37)(18,24,26,28,27,40,36,39)(20,29,34,47,35,46,33,45)(25,38,42,43,41,44,32,48);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,25)(19,29)(21,34)(22,33)(23,35)(24,38)(26,42)(27,41)(28,43)(30,47)(31,46)(32,36)(37,45)(39,48)(40,44);(2,8)(6,14)(7,9)(13,15)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=gi,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, 107924);