#8108721

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	#59807	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4504 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015917);