#8284466

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8191662);