#8436283

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

4412 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=h,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, 8343479);