#6438655

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×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3176 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^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, 6345851);