#6573825

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	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2603 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,23)(22,39)(24,31)(25,34)(26,33)(27,36)(28,37)(29,32)(30,38)(35,40);(1,2,3,4)(9,11)(10,16)(12,15)(13,25)(14,23)(17,32)(18,35)(19,21)(20,31)(22,28)(24,29)(26,30)(27,40)(33,37)(34,36)(38,39);(6,8)(11,12)(15,29)(16,31)(17,21)(20,32)(22,25)(23,24)(26,40)(28,30)(33,35)(34,39)(37,38);(9,12,14,24)(10,17,19,31)(11,15,23,29)(13,26,27,22)(16,32,21,20)(18,33,36,39)(25,30,40,28)(34,38,35,37);(9,13,14,27)(10,18,19,36)(11,22,23,26)(12,25,24,40)(15,30,29,28)(16,33,21,39)(17,34,31,35)(20,38,32,37)

Polycyclic presentation:

2^9|b^2=fi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,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, 6481021);