#8289263

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

4454 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=ghi,e^2=gh,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, 8196459);