#8302119

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	#40515	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4349 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=ghi,f^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, 8209315);