#8463237

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4246 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=gh,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8370433);