#6444207

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,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, 6351403);