#8276601

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

4628 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)(3,7,8,5)(9,10)(11,17)(12,20)(13,15)(14,19)(16,22)(18,23)(21,24)(25,26,31,34)(27,35,36,32)(28,33,37,40)(29,38,39,30);(1,3,4,8)(2,5,6,7)(9,11,13,19)(10,14,15,17)(12,18,22,21)(16,23,20,24)(27,36)(28,37)(32,35)(33,40);(1,3)(2,5)(4,8)(6,7)(10,15)(14,17)(16,20)(23,24)(25,27,31,36)(26,32,34,35)(28,29,37,39)(30,40,38,33);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,21)(13,19)(15,17)(16,24)(18,22)(20,23)(25,28)(26,33)(27,29)(30,32)(31,37)(34,40)(35,38)(36,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,22)(14,23)(15,20)(17,24)(19,21)(25,29)(26,30)(27,37)(28,36)(31,39)(32,40)(33,35)(34,38);(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)(25,30)(26,29)(27,33)(28,32)(31,38)(34,39)(35,37)(36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,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, 8183797);