#206390

order	512
nilpotency class	3
isoclinism	isoclinic class of #19697
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×#127
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

953 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=fi,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=di

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, 113586);