#8583455

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	#35361	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4580 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8490651);