#8201640

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	#55034	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4543 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)(3,5,7,8)(9,10)(11,19)(12,21)(13,18)(14,16)(15,22)(17,23)(20,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(3,7)(5,8)(9,11,14,22)(10,15,16,19)(12,18,23,20)(13,21,24,17)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,2)(3,5)(4,6)(7,8)(10,16)(13,24)(15,19)(17,21)(26,30)(31,33)(32,36)(38,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,23)(10,17,16,21)(11,20,22,18)(13,15,24,19)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,13,14,24)(10,18,16,20)(11,21,22,17)(12,15,23,19)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40)

Polycyclic presentation:

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

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, 8108836);