#8591891

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	#40723	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4469 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=gh,e^2=g,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, 8499087);