#8678985

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4279 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8586181);