#8509932

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4334 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)(11,16)(12,21)(13,17)(14,20)(15,19)(18,24)(22,23)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,19,23)(12,18,13,20)(14,21,24,17)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(2,5)(6,8)(9,12,15,13)(10,17,19,21)(11,20,22,18)(14,16,24,23)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(3,7)(6,8)(9,13,15,12)(10,17,19,21)(11,18,22,20)(14,16,24,23)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,18)(11,21)(12,23)(13,16)(15,24)(17,22)(19,20)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,21)(14,18)(15,19)(20,24)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8417128);