#8196370

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	#36440	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4508 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(25,26)(27,28)(29,32)(30,38)(31,34)(33,39)(35,37)(36,40);(1,2,4,6)(3,5,8,7)(10,15)(11,18)(17,20)(19,24)(25,27,31,32)(26,28,34,29)(30,35,39,40)(33,36,38,37);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,28,31,29)(26,32,34,27)(30,36,39,37)(33,40,38,35);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,29,31,28)(26,32,34,27)(30,37,39,36)(33,40,38,35);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,30,31,39)(26,33,34,38)(27,35,32,40)(28,36,29,37);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,40)(38,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103566);