#8199326

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

4336 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=gh,e^2=h,f^2=g,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, 8106522);