#200879

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1097 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,f^2=h,g^2=hi,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, 108075);