#8436224

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

4376 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,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, 8343420);