#6438677

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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

3153 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)(9,10,14,17)(11,15,21,31)(12,16,25,33)(13,18,26,35)(19,28,22,30)(20,29,37,39)(23,32,38,40)(24,34,27,36);(5,6,8,7)(9,11)(10,15)(12,23)(13,19)(14,21)(16,32)(17,31)(18,28)(20,24)(22,26)(25,38)(27,37)(29,34)(30,35)(33,40)(36,39);(2,4)(5,7,8,6)(11,19)(13,26)(15,28)(18,35)(20,23)(21,22)(24,27)(29,32)(30,31)(34,36)(37,38)(39,40);(9,12,14,25)(10,16,17,33)(11,20,21,37)(13,27,26,24)(15,29,31,39)(18,36,35,34)(19,23,22,38)(28,32,30,40);(10,17)(11,21)(16,33)(18,35)(19,22)(20,37)(23,38)(34,36)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^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, 6345873);