#8127478

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

4406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8034674);