#8509485

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,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, 8416681);