#8525820

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8433016);