#6438687

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	C₂×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3226 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:

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

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,c^b=ch,d^b=dh,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, 6345883);