#200820

order	512
nilpotency class	3
isoclinism	isoclinic class of #200737
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1041 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=hi,e^2=hi,f^2=h,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=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, 108016);