#8600854

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

4442 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=h,e^2=g,f^2=gh,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, 8508050);