#6448027

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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,h^b=hi,h^c=hi,h^d=hi

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, 6355223);