#201569

order	512
nilpotency class	3
isoclinism	isoclinic class of #201404
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1161 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)(17,18,22,27,23,26,21,37)(19,28,32,39,31,43,30,24)(20,33,35,41,36,25,34,40)(29,42,46,38,45,48,44,47);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19)(18,24,26,39)(20,29)(21,30)(22,32)(23,31)(25,38,33,47)(27,28,37,43)(34,44)(35,46)(36,45)(40,48,41,42);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,20,23,36)(18,25,26,33)(19,29,31,45)(21,35,22,34)(24,38,39,47)(27,41,37,40)(28,42,43,48)(30,46,32,44);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(18,26)(24,39)(25,33)(27,37)(28,43)(38,47)(40,41)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108765);