#8599582

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

4605 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506778);