#6574969

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	#61057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1991 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)(11,23)(12,27)(13,20)(14,21)(15,18)(16,22)(17,26)(19,24)(25,39)(28,36)(29,35)(30,37)(31,34)(32,38)(33,40);(2,5)(6,8)(9,11,14,26)(10,17,21,23)(12,16,24,15)(13,29,30,28)(18,19,22,27)(20,36,37,35)(25,32,40,31)(33,38,39,34);(1,3,4,7)(2,6,5,8)(10,18)(11,24)(12,26)(15,16)(19,27)(20,34)(21,22)(25,29)(28,40)(31,32)(35,36)(37,38);(9,12)(10,19)(11,16)(13,25)(14,24)(15,26)(17,22)(18,23)(20,33)(21,27)(28,31)(29,32)(30,40)(34,35)(36,38)(37,39);(9,13,14,30)(10,20,21,37)(11,25,26,40)(12,28,24,29)(15,31,16,32)(17,33,23,39)(18,34,22,38)(19,35,27,36)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,e^2=i,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, 6482165);