#8131484

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4466 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=ghi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8038680);