#8197804

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

4643 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)(11,19)(12,21)(13,18)(14,16)(15,22)(17,23)(20,24)(25,26,31,29)(27,34,30,32)(28,36,38,33)(35,39,37,40);(1,2)(3,7)(4,5)(6,8)(9,11,14,22)(10,15,16,19)(12,18,23,20)(13,21,24,17)(25,27,31,30)(26,32,29,34)(28,35,38,37)(33,39,36,40);(2,5)(6,7)(10,16)(13,24)(15,19)(17,21)(25,26,31,29)(27,32,30,34)(28,33,38,36)(35,39,37,40);(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,28)(26,33)(27,35)(29,36)(30,37)(31,38)(32,39)(34,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,23)(10,17,16,21)(11,20,22,18)(13,15,24,19)(25,29,31,26)(27,32,30,34)(28,36,38,33)(35,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,24)(10,18,16,20)(11,21,22,17)(12,15,23,19)(25,30,31,27)(26,32,29,34)(28,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,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^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, 8105000);