#8188179

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

4448 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)(4,6)(5,8)(9,10)(11,15)(12,17)(13,21)(14,16)(18,22)(19,20)(23,24)(25,26)(27,33)(28,29)(30,34)(31,32)(35,40)(36,37)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,23,21,20)(26,32)(28,33)(34,39)(36,40);(1,2,4,6)(3,5,8,7)(10,16)(11,17)(18,21)(19,24)(25,27,31,29)(26,33,32,28)(30,37,38,35)(34,36,39,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,15)(10,17,16,11)(13,20,22,23)(18,24,21,19)(25,28)(26,27)(29,32)(30,36)(31,33)(34,35)(37,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,13,14,22)(10,18,16,21)(11,19,17,24)(12,20,15,23)(25,29,31,27)(26,28,32,33)(30,37,38,35)(34,36,39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,24)(21,22)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=i,e^2=gh,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, 8095375);