#8633918

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

4411 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8541114);