#200607

order	512
nilpotency class	3
isoclinism	isoclinic class of #200430
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1026 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,22,28,23,26,21,27)(19,29,33,24,32,40,31,39)(20,34,36,41,37,25,35,42)(30,43,46,47,45,48,44,38);(1,3,4,11,6,13,5,12)(2,7,14,16,9,15,8,10)(17,19)(18,24)(20,30)(21,33)(22,31)(23,32)(25,38)(26,39)(27,40)(28,29)(34,47)(35,46)(36,44)(37,45)(41,43)(42,48);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(17,20,23,37)(18,25,26,34)(19,30,32,45)(21,36,22,35)(24,38,39,47)(27,42,28,41)(29,43,40,48)(31,46,33,44);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(18,26)(24,39)(25,34)(27,28)(29,40)(38,47)(41,42)(43,48)

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,e^b=ei,f^b=fi,g^b=gi,d^c=dh,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, 107803);