#205589

order	512
nilpotency class	3
isoclinism	isoclinic class of #9758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

941 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112785);