#8246163

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

4381 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8153359);