#8436269

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
prime radical	#40669	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4358 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)(10,14)(11,17)(16,22)(18,19)(26,31)(27,34)(28,36)(37,39);(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(1,2)(3,6)(4,5)(7,8)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,3,4,7)(2,6,5,8)(9,11,13,17)(10,15,14,21)(12,19,20,18)(16,24,22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gh,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, 8343465);