#204441

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=eh,g^b=gh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111637);