#8371313

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

4446 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8278509);