#8581769

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

4711 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488965);