#8119561

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

4423 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026757);