#8599446

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	#38512	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4385 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,e^2=gh,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, 8506642);