#6574961

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61055	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1983 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,28)(13,30)(14,18)(15,20)(16,25)(17,31)(19,34)(22,39)(23,29)(24,27)(26,35)(32,33)(36,40)(37,38);(2,5)(6,8)(9,11)(10,16)(12,29)(13,23)(14,25)(15,26)(17,36)(18,21)(19,35)(20,24)(22,31)(27,34)(28,37)(30,38)(32,40)(33,39);(1,3,4,7)(2,6,5,8)(9,12,14,30)(10,17,18,33)(11,22,25,40)(13,20,28,19)(15,32,34,31)(16,23,21,37)(24,36,35,39)(26,29,27,38);(9,13)(10,12)(11,23)(14,28)(15,17)(16,29)(18,30)(19,31)(20,32)(21,38)(22,35)(24,40)(25,37)(26,36)(27,39)(33,34);(9,10)(11,24)(12,17)(13,32)(14,18)(15,19)(16,26)(20,34)(21,27)(22,36)(23,29)(25,35)(28,31)(30,33)(37,38)(39,40)

Polycyclic presentation:

2^9|c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,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, 6482157);