#6448049

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355245);