#8346183

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

4466 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:

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

Polycyclic presentation:

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

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, 8253379);