#8532124

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=i,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8439320);