#6444159

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂×#3041	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,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, 6351355);