#8433734

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

4712 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)(6,8)(9,10,16,15)(11,20,14,17)(12,19,13,18)(21,23,22,24)(25,26,32,31)(27,36,38,33)(28,34,29,39)(30,40,37,35);(3,7)(6,8)(9,11,16,14)(10,17,15,20)(12,22,13,21)(18,24,19,23)(25,27)(26,33)(28,30)(29,37)(31,36)(32,38)(34,35)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,15,19)(11,21,14,22)(17,23,20,24)(25,28,32,29)(26,34,31,39)(27,30,38,37)(33,35,36,40);(1,3)(2,6)(4,7)(5,8)(9,13,16,12)(10,19,15,18)(11,22,14,21)(17,24,20,23)(25,29,32,28)(26,34,31,39)(27,37,38,30)(33,35,36,40);(1,2)(3,6)(4,5)(7,8)(9,14,16,11)(10,20,15,17)(12,22,13,21)(18,24,19,23)(25,30,32,37)(26,35,31,40)(27,28,38,29)(33,34,36,39);(9,15,16,10)(11,20,14,17)(12,19,13,18)(21,24,22,23)(25,31,32,26)(27,36,38,33)(28,39,29,34)(30,40,37,35)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8340930);