#8621894

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	#46378	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4370 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,e^2=i,f^2=h,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, 8529090);