#6573834

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#36020	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2601 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)(7,8)(9,10,13,17)(11,19,22,15)(12,16,26,35)(14,18,28,37)(20,39,40,30)(21,32,23,31)(24,34,25,33)(27,36,29,38);(1,2,3,4)(9,11,13,22)(10,15,17,19)(12,24,26,25)(14,21,28,23)(16,33,35,34)(18,31,37,32)(20,29,40,27)(30,38,39,36);(6,8)(9,12,13,26)(10,16,17,35)(11,20,22,40)(14,29,28,27)(15,30,19,39)(18,38,37,36)(21,24,23,25)(31,33,32,34);(9,11,13,22)(10,15,17,19)(12,25,26,24)(14,21,28,23)(16,34,35,33)(18,31,37,32)(20,27,40,29)(30,36,39,38);(11,21)(14,28)(15,31)(18,37)(19,32)(20,24)(22,23)(25,40)(27,29)(30,33)(34,39)(36,38)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481030);