#8188122

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4402 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24)(26,28)(27,33)(32,36)(35,39);(1,2)(3,6)(4,5)(7,8)(10,11)(15,17)(16,18)(22,23)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(2,5)(6,7)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095318);