#8343549

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	#59677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4324 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=h,e^2=gi,f^2=gi,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, 8250745);